SpongeBob "I Fucking Love X" - It was a very Specific Mood

If "Zoomer Shooter" were a genre, it should refer to stuff like Fortnite, not Halo. Not to mention that even in the 90's, shooters came in all sorts of styles. Rainbow Six was a fairly tactical multiplayer game, Doom was a fantastic one man power fantasy, Half-Life was a story-focused experience, etc.

If I had to group Gears of War, Halo, Mass Effect, and Fallout 3 together, I'd emphasize how they are somewhat more fantastic than realistic, have sci-fi settings, they are largely defined by their story-based campaigns, and were most popular on Xbox 360 (depending on the context, the omission of popular PC games from the era like Team Fortress 2, Crysis, etc).

Maybe something like "Millenial Sci-Fi Shooter?"

And then there's Spec Ops: The Line or Singularity