Culturally, how long did each decade last for gaming?

Yeah, the fact the PS2 could play DVDs certainly helped it become popular, but that doesn't really have anything to do with video games. There were non-gamers who bought a PS2 just so they could watch movies on it.
But as far as gamers were concerned, the 6th gen really took off in late 2001.

I would say these school years are how the transition from the 3rd to 9th gen would be like:

1989-90' = transitional with the 4th gen because of the Genesis release but the NES was still in its prime so predominantly 3rd gen.

1990-91' = transitional leaning 3rd gen as the NES still had relevance but the Genesis was really getting popular. 

1991-92' = first core 4th gen year with the release of the SNES and Genesis really taking over with the release of Sonic. NES was probably still relevant but the SNES releasing really overshadowed it.

1992-93' = the absolute peak of the 4th gen with the many notable titles that would come out on the Genesis and SNES.

1993-94' = last core 4th gen year before the hype of the 5th gen would come about.

1994-95' = predominantly 4th gen but transitional to the 5th gen due to the PS1 release in Japan and the Saturn release.

1995-96' = 50/50 but might lean towards 4th gen because of the PS1 USA release but most people probably still had a Genesis or SNES.

1996-97' = predominantly 5th gen but transitional to the 4th gen due to the N64 release and PS1 gaining more popularity as more households own it.

1997-98' = first core 5th gen year as the main 5th gen consoles like PS1 and N64 truly took off and had their more groundbreaking titles. Probably the last school year with any relevant influence from the 4th gen though.

1998-99' = the absolute peak of the 5th gen as the PS1 and N64 had their most quintessential titles throughout this gaming season, as well as the Gameboy Color releasing this school year, and the old-school 4th gen consoles that defined the classic 90's were all discontinued but before the hype of the 6th gen, despite the Dreamcast releasing in Japan.

1999-00' = debatably a core 5th gen year but because of the massive influence and rise (and essentially the peak) of the Sega Dreamcast as it released this school year, this was the first sorta transitional school year into the 6th gen. Despite that, the 5th gen totally dominated.

2000-01' = this was the true transitional year for the 5th gen and 6th gen as the Dreamcast was still in its peak but ended up becoming discontinued (the ultimate Y2K console) and the N64 and PS1 were still in its prime (but was the last school year of its prime) and the PS2 (the first true 6th gen console) released and had massive popularity but it was still overshadowed by the PlayStation and Nintendo 64, so it was more of a 5th gen year.

2001-02' = transitional with the 5th gen as it was still popular and a lot of kids still had N64s and PS1s, however the 6th gen truly blew up this school year with many groundbreaking titles for the generation, as well as their other consoles, GameCube and XBOX, releasing that fall, and despite its discontinuation, the Dreamcast still had relevance here. It's more of a 6th gen year by far however could debatably be a core 6th gen year though

2002-03' = first core 6th gen year as the generation fully took over and reached new heights as it gained online capabilities like XBOX Live and PS2 getting online support, and the titles that came out this year really put the 6th gen into a zeitgeist. This was the last school year however that the 5th gen had any real relevance, especially since the N64 would discontinue mid-way that school year.

2003-04' = this was without a doubt the absolute peak of the 6th gen as the 5th gen influence was insignificant (despite the PS1 still being around unitl 2005/06) and there was no real hype for the 7th gen yet. So many notable titles that would define this era of video games.

2004-05' = last core 6th gen year as this was the last true essence of what the 6th gen had to offer but the first technical 7th gen consoles (despite all being handheld), the Nintendo DS and PSP came out, which would cause some hype for the incoming 7th generation of video game consoles, especially since the Microsoft and Sony would reveal their new generation consoles around E3 2005. This year probably had the most notable 6th gen titles or at the very least the quintessential titles of the 2000's.

2005-06' = overall a predominate 6th gen year as there would still be titles that would define the generation but clearly transitional as the first true 7th gen console the Xbox 360 came out, and the 6th gen did not have as much of the excitement as it did in the previous school years.

2006-07' = the 50/50 period as the 6th gen was still dominant and relevant but the 7th gen was starting to take over as the remaining consoles, the PS3 and Wii, released that year. Because of that, this might lean towards the 7th gen.

2007-08' = the last sorta transitional school year as the PS2 was still very relevant however the XBOX was irrelevant at this point and the GameCube got discontinued. The PS3 was not very popular because of its lack of titles and expensive price and hardware failures (this was a low point for Sony around this time), however the Wii was at its ultimate peak and the Xbox 360 was also peaking. Debatably a core 7th gen year.

2008-09' = first core 7th gen year as all three 7th gen consoles took over and most households had it, as well as many notable 7th gen titles releasing. Probably the last real influence from the 6th gen as the XBOX was discontinued this school year (PS2 didn't get discontinued until 2013 but still).

2009-10' = peak 7th gen year as all three 7th gen consoles were doing pretty good since PS3 had a renaissance period and was starting to take over but was still at even ground with the 360 and Wii.

2010-11' = about the same but the PS3 was at its apex and was gaining more popularity while the Wii was getting less relevant. The 3DS would release though which was probably the first sign of the 8th gen.

2011-12' = pretty much the same but the PS Vita would come out this year and this was probably the beginning of the 8th gen hype most likely, if not, then definitely by the next school year.

2012-13' = debatably the last core 7th gen year as the PS3 and 360 were still totally dominating, however could be a transitional year because of the first real 8th gen console, the Wii U, releasing and the PS4 and Xbox One being revealed in early 2013 but in terms of relevance, the Wii U releasing didn't really change much and it still felt like a core 7th gen year so I lean toward the former.

2013-14' = the first true transitional year as the PS4 and Xbox One released for the 8th gen, however most people still had a 7th gen console, so contrary to popular belief, this leans toward 7th gen.

2014-15' = the last true transitional year as more people were getting the new 8th gen consoles and the 8th gen were starting to get more games as they kinda lacked titles in the previous school year and it was less affordable then but started getting more affordable this school year but the 7th gen still had relevance.

2015-16' = the very last potential transitional year as this was the last school year where the 7th gen still seemed slightly relevant and current without feeling dated, but is probably moreso a core 8th gen year, especially by the holiday season of 2015 as that's when my friends and cousins got their PS4s and when you noticeably start seeing more and more people own an 8th gen console, and it doesn't help that the Xbox 360, the first 7th gen console turned 10 that year, making the 7th gen age more than it should.

2016-17' = the absolute peak of the 8th gen as that's when the PS4 and Xbox One totally dominated and had more notable titles and released their slim and pro versions, as well as the Nintendo Switch coming out, which pretty much kickstarted the second wave of the 8th gen as the Wii U became totally irrelevant at this point, which was a part of the first wave of the 8th gen.

2017-18' = same as 2016-17'

2018-19' = same as the previous school years

2019-20' = undoubtedly the last REAL core 8th gen year however this was the beginning of the 9th gen hype as the PS5 was finally revealed to the public after years of questioning whether there would actually be a PS5 at all.

2020-21' = transitional but clearly more 8th gen as the 9th gen consoles were not getting many sales and easily selling out so most people stuck to their 8th gen consoles by force and couldn't afford the new ones.

2021-22' = 50/50 as a lot of people still have 8th gen consoles and it is still very relevant but the push to 9th gen really surged here and more people have been getting the new gen consoles

2022-23' = might still be transitional but will lean 9th gen as more people buy the newer consoles.

Halo took a few months to become super popular, around February 2002 it exploded, was fun witnessing it

This is how I'd rank them

1989-1990: Peak 3rd gen. This school year is defined by The Wizard and Super Mario Bros 3.
1990-1991: Still more 3rd gen, but transitioning to the 4th gen as the Genesis is becoming popular due to superior graphics and capabilities.
1991-1992: 4th gen dominant. SNES is released whilst Sonic the Hedgehog results in the Genesis really exploding in popularity. The console wars really ignite.
1992-1993: Same as 1991-1992. Sonic is even more popular now, and Sega, in general, is in its prime.
1993-1994: The peak of the console wars and the 16-bit era in general. Mortal Kombat's censorship on the SNES results in increased popularity of the Genesis coupled with the release of Sonic 3. The absolute peak of the Genesis.
1994-1995: Last school year the Genesis is popular, whilst the SNES rapidly grows in popularity due to landmark titles, notably Donkey Kong Country. The Saturn is released but does not become popular, resulting in Sega's decline. IMO the absolute peak of 90s gaming in general.
1995-1996: Last 16-bit/4th gen dominant school year, as the PlayStation is released. However, the SNES is still very popular, thanks to the likes of Donkey Kong Country 2, Yoshi's Island, and Super Mario RPG, which are all acclaimed.
1996-1997: 3D gaming takes over with the release of the Nintendo 64, and landmark titles like Super Mario 64, Crash Bandicoot, and Tomb Raider are released. Donkey Kong Country 3 is the final hurrah of the 16-bit era, but does not gain as much attention or praise as earlier titles, especially with new 3D platformers.
1997-1998: 16-bit gaming is now irrelevant, and 5th gen gaming is dominant. Final Fantasy VII increases sales of the PS1, helping it become the best-selling system of the generation.
1998-1999: The peak of the 5th gen, with landmark titles releasing on both the N64 and PS1, not to mention the release of the Game Boy Color. Also often regarded as one of the greatest years in gaming history.
1999-2000: The Dreamcast debuts, but gaming is still very much dominated by the PS1. This is where the transition between the 90s and 00s begins.
2000-2001: Last school year 5th gen gaming is dominant. Sega exits the console market and discontinues the Dreamcast, whilst the PlayStation 2 is released. Despite this, this school year is still defined more by PS1 and N64 games.
2001-2002: A landmark school year, which sees the PS2 truly taking off, the GameCube, Game Boy Advance, and Xbox launching. The 2000s in gaming truly begins here.
2002-2003: Same as before, but now online gaming has arrived with Xbox Live and PS2 Online. The Game Boy Advance SP is released and is a major update to the GBA (introducing the long-awaited backlight and with a flip design that would anticipate the Nintendo DS).
2003-2004: Same as 2002-2003.
2004-2005: Nintendo DS and PSP are released, but this is still a Game Boy Advance dominant school year. Halo 2 helps popularize Xbox Live. The absolute peak of 2000s gaming in my opinion.
2005-2006: The Xbox 360 is released, but this is the last school year that is really dominated by the 6th gen. As I remember it, the OG Xbox was still more popular than the Xbox 360 at this time. The Nintendo DS and PSP take off during this school year, but the GBA still has some relevance.
2006-2007: 7th gen gaming really takes off, with the release of the Wii and PS3, and the Xbox 360 really takes off with notable titles like Gears of War. The 6th gen still has some popularity, and there are still big titles being released like Twilight Princess (also a Wii title) and God of War II, giving it more relevance than it would have in 2007-2008. In terms of handhelds, this is the first 7th gen dominant school year.
2007-2008: The Xbox 360 and Wii are in their prime, whilst the PS3 was not popular and in fact was mocked a lot online during this period. The first 7th gen dominant school year, and also like 1998-1999, is regarded as one of the best years in gaming history.
2008-2009: Same as 2007-2008, except the Nintendo DSi releases.
2009-2010: PS3 really takes off with the release of the PS3 Slim and notable games like Uncharted 2 and Heavy Rain. Coupled with the Wii still being popular, and this makes it the peak 7th gen school year. The transition to the 2010s starts in this school year.
2010-2011: Last school year the Wii was truly popular in (Super Mario Galaxy 2, Sonic Colors, Skyward Sword), whilst also seeing the release of the Kinect. The Nintendo 3DS is released.
2011-2012: By this point, the Wii is no longer popular and Nintendo begins to enter its dark age. The PlayStation Vita is released.
2012-2013: The Wii U is released, whilst the PS3, once unpopular and mocked, outsells the Xbox 360 at the end of the year. This is the first really 2010s feeling school year IMO.
2013-2014: The PlayStation 4 and Xbox One are released. Despite this, I'd still say this school year leans more toward the 7th gen, with GTA V for example initially releasing for PS3 and Xbox 360 first for example.
2014-2015: This school year leans more 8th gen, but is ultimately still transitional between the 7th and 8th gen. As I remember it distinctly, many games during this time were released for both generations.
2015-2016: The first truly 8th gen dominant school year. The 7th gen still had some relevance but this was the last school year it had any relevance.
2016-2017: The 7th gen is now dead by this point. The Nintendo Switch is released, replacing both the Wii U AND 3DS. This fact, coupled with the fact the PS4 was doing incredibly well, with the PS4 Pro releasing this year, makes this IMO the peak of the 8th gen. I also regard it as the definitive 2010s school year for gaming.
2017-2018: Same as 2016-2017 except Microsoft releases their own updated version of their console, the Xbox One X.
2018-2019: Same as 2017-2018.
2019-2020: Same as before.
2020-2021: The launch of the PS5 and Xbox Series X launches the 9th gen, however this school year is still dominated by the 8th gen. The shift to the 2020s started here.
2021-2022: This school year is more 9th gen, but ultimately still transitional, especially with the Switch still being a success and the fact new games are still being released for the Xbox One and PS4 simultaneously with the Xbox Series X and PS5.