Well game genres are rarely defined by the strict definition of their name. Role-playing games rarely have you actually play a role, they're defined as levels and stats and items you can pick up from loot crates. You can play an arcade game on a home console or PC, no quarters required. I play card games on my phone without actually owning a deck of cards.

An FPS has to be a shooter that's in first-person view, but does that make every game where you shoot something with a first-person camera view an FPS? What if the game has other elements? Which definition wins out? Is Fallout more RPG, or is it more FPS? Or is it a turn-based strategy game? Most people would say it's an RPG that happens to have guns and is played in first-person mode and part of the game mechanics is reminiscent of turn-based games.

When you say "FPS", it's a pretty strict definition, because if it's an RPG that happens to be played in first-person with a gun, it's going to be called an RPG first. Maybe RPG-FPS. But FPS really means "run around with a gun and your main interaction with the world is shooting it". There is an entire world of games that do exactly that, and we call them FPS games. Borderlands is an FPS with hints of MMO, dungeon crawler, and RPG, but since the main interaction with the world is running around with a gun and shooting things, it's an FPS. I don't think that's a bad thing.