Also I question whether "FPS" is an actual genre, or just a 3D version of "shooter" -- a lot of places group shmup and FPSs together, even though they play very differently, as "shooters". It's kind of like calling isometric games a genre -- a visual perspective doesn't itself a genre make. Doom-clone is at least a subgenre though.
If you question that, then I question that "shooter" is even a genre, its just a subgenre of "action" games, a lot of places group platformers and shooters together, even though they play very differently, as "action" games. It's kinda like calling 3D Platformer games a genre -- visual representation doesn't itself a genre make. Bullet-Hell is at least a subgenre though.
Still, my point is, this ratchets up and down, I look at it like an evolutionary tree, kinda like how "A Theory of Fun" tracked the progression of space invaders clones, at any point in that tree you can take a slice and define genres based on features and play styles. Right now I think 3D shooters are a Genre and within that Genre, there are a multitude of subgenres and to ignore the amount of different yet similar games is doing it a disservice.