I started a bit earlier.

Wolfenstein 3D
Doom
Duke Nukem 3D (played tons in LAN multiplayer)
Rise of the Triad (just bonkers)
Half-Life / Half-Life Death match
Quake
Unreal / Unreal Tournament
Halo
Far Cry
Half-Life 2

I never really got into online multiplayer, too spoiled with LAN multiplayer. We always played Duke Nukem 3D, HL Death match or Quake at the end of the work day at our software company. One of us even made a HL level of the office building so we could shoot up / duke it out in the office at the end to the day. I played Unreal Tournament at home a lot with 3 PCs, Halo on split-screen.

Fast movement plus mouse look is still my favorite. By head look and free aiming (Aeon of Ruin VR, basically Doom in VR) works even better.

2004-2010. Unreal Tournament 2004, Halo 2, 3, Reach. CoD 4, WAW, MW2, Blackops. Left 4 Dead 1+2, Battlefield Bad Company 1+2, Killzone 2, Bioshock, Half-Life 2

For multiplayer I really miss arena shooters. And among the arena shooters UT99 rules them all!

Absolutely. 100%

Every weekend now, I'm playing local Quake 1 multiplayer and wow... what a game.

Well said. I would say 1996-2005 was the golden age.

N64 knocked it out of the park with GoldenEye 007, Perfect Dark, and the Turok series. PlayStation held its own with Medal of Honor 1 & 2 and Alien Trilogy. On PC, we got the Unreal series (including the underrated first game), the Quake series, Doom 3, Half-Life 1 & 2, Counter-Strike, Return to Castle Wolfenstein, Call of Duty 1 & 2, F.E.A.R., and many more. PlayStation 2/Xbox/GCN together gave us Killzone, Red Faction, the TimeSplitters trilogy, Nightfire, Halo 1 & 2, Republic Commando, etc.

And that excludes borderline FPS games like Metroid Prime and Escape from Butcher Bay.

Half-Life 1-2 (and Black Mesa 20 years after).

As for multiplayer I legitimately think Overwatch was incredible at launch, that and Modern Warfare 1-2 for very different reasons.

Oh and BTW when talking UT99 (and UT2003/2004, Q3 and Q1):

You young ones that say games are unplayable at 30 FPS, you can have your 120+ FPS I’ll still kick your ass at 25 FPS!

Oh man, I forgot that Crysis was that year too.  That just strengthens my stance even more.  I didn't see it until later on a friend's PC so it's often overlooked by me as one of the many incredible games of 2007 (Galaxy, God of War 2, Uncharted, Portal, AC, Prime 3, etc).

Just because we talked about it I installed UT99 right now!

Everything is available right here https://oldunreal.com/downloads/unrealtournament/full-game-installers/

A bonus for me that is running a Mac is that the community patch is native to Apple Silicon

Gen 7.

They didn't call the Xbox360 the Shooter box for nothing.