I've seen some FPSes get the player's preferred control mappings in a really subtle, cool way: ask the player to perform some actions very early in the game (often as part of a "training" sequence), and configure the controls however necessary to make the user's input correct.
For instance, in Portal 2, to check whether the player wants inverted controls:
I'm guessing whichever direction your move the mouse, the game looks in the correct direction for you to obey the instructions, and that mapping is then persisted.
It makes more sense for consoles / controllers. Although I am sure /someone/ uses inverted mouse controls. It could be disabled in the PC version also.
Inverted Y (vertical axis, i.e. up/down view) is a very popular setting for simulation games and FPS games on PC. It's popular enough that games with >100k players will get bad press and >20 forum complaints (even when the forum is hard to find!) over it, and many more than this will simply not play/buy the game.
In other genres / types of games, especially where you don't control exclusively one character / vehicle from a fixed first/third person viewpoint, it is drastically less common.