Street Fighter 4: Here's a weird one. You know the presentation of your multiplayer tag or badge or whatever the fuck they call it in Call of Duty that other games do too? That's from Street Fighter 4. I'd say that was probably SF4's biggest influence. Other than that, I think a little too much is made of SF4 bring fighting games back, specifically this would mean fighting games on a 2D plane since the 3D ones were still doing alright. Like there were still 2D games at the time, (I'd even say BlazBlue and KoF12 were much better fighting game that SF4, especially that first version with the wonky feeling jump) and Capcom and Marvel were already talking about MvC3 before Street Fighter 4 even came out. I can't remember his name, but the new at the time head of Marvel's gaming devision accidently let it slip at one of the trade shows showing SF4 before release that they were doing a new Marvel game; although because of the timetable this retroactively got chalked up to being a leak about the 2009 MvC2 rerelease. It does however seem the success Capcom had, the the criticism of Mortal Kombat vs DC Universe lead to Mortal Kombat returning to being a 2D fighting game.
Grand Theft Auto 3: I mean, this is an easy one. You can also point to the other game they made why still called DMA design: Body Harvest. But GTA3 is the reason we've got all these other openworld sandbox games.
Saints Row 2 - 4: The presentational wackiness of characters in Fortnite always felt like an extention of the wackiness of Saints Row and what people did with that games create a character and how you could dress your character.
Splinter Cell and Resident Evil 4: The over the shoulder third person camera. Splinter Cell did it first, but RE4 seems to be the reason you see it in everything. Why it took RE4 for this particular aspect of Splinter Cell to catch on I've no...maybe because RE4 was an action game and Splinter Cell (despite having action) was mostly seen as a stealth game. Funnily Splinter Cell also has something like this happen with it's cover system. WinBack was the first game to do cover like Splinter Cell does, but it wasn't until Splinter Cell that you started seeing it.
Kill.Switch: Kill Switch is specifically noted as a influence for Gears of War's blind fire.
Def Jam: Fight for NY: This game, although probably more specifically AKI's wrestling engine in general seems to have been an influence on a wave of similar 3D beat 'em ups and fighting game that followed it for a few years. Of these games, only Yakuza went on to be an ongoing series; although apparently the Shadow of Rome sequel ended up morphing into Dead Rising.
Devil May Cry: The influence of DMC is somewhat odd, because is highly influential, clearly games like God of War and Heavenly Sword are pulling from this. At the same time, those games feel very different, and the only other game to really feel like Devil May Cry is Bayonetta...which was made by the creator of DMC. I will see Devil May Cry attacks in other games from time to time too.
Anarchy Reigns: Overwatch completely ripped off this game's cast of characters for their original lineup of characters in that game.
Geometry Wars: This game kicks off Xbox Live Arcade being this force for smaller games from companies and eventually indie games. For a long time Xbox Live was the home of this, then Steam around like 2012 with Hotline Miami and Steam Workshop, and now it seems to be the Switch. Probably also why you started seeing more retro looking games, although the initial retro looking (like Geometry Wars) games visually had a modern spin with stuff like Pac-Man Championship Edition, Galaga Legions, Space Giraffe, Space Invaders Extreme, and Braid.
Cave Story: More it's self publishing model than the game itself. Although I'd guess it's also why you see so many indie games that are some kind of take on Metroid and Castlevania.
Spelunky: The reason there's all these god damn Rogue-lite games now. It is kind of interesting to think about how long it's taken for elements of Rogue to drift into western games given its popularity in Japan with the Mystery Dungeon games. Until this post Spelunky period of games like FTL, Binding of Isaac, and Rogue Legacy the most influence Rogue seemed to have was random level generation in Diablo.