Crazy sales and millions of console FPS players would disagree. I think playing an FPS on a controller is "normal" and just fine as of the current generation of consoles. Of course there will be edge cases, too (quake 3, CS, etc.).
So FPS games have changed dramatically to work on consoles. They are often third person, and it gives you a better sense of space on a display so far away. They have become 'slower' and the fighting has become mostly horizontal and far less vertical, because controllers just aren't as good as a mouse in terms of aiming. The games tend to have lots of 'shooting' gallery style situations since it's harder to both move and shoot. The whole chest-high wall thing is another by-product, as is auto-aim.
I'm not saying you can't play FPS: you definitely can, but a mouse fits better. I find console FPS / TPS games are easier on PC because the mouse aiming is just so much easier.
I can't think of a way of phrasing this in a way that doesn't sound like "my dad could beat up your dad", so I'll simply say it and hope for the best: I'd wager than an average FPS player on a mouse + keyboard would be 'better' than a good FPS player using a controller, due simply to mouse superiority.
(I haven't played a console FPS in a couple of years though, perhaps things have changed)
Ironically, Battlefield 3 has sold roughly 2M PC copies, 6M Xbox copies, and 5M PS3 copies. [1,2,3] It was numbers like these that caused most of the game franchises to switch to console if they weren't already focused on them. A lot started out being PC-focused such as Ghost Recon, Call of Duty, Splinter Cell and The Elder Scrolls come to mind, but all of their latest franchise titles are heavily designed as console games first.
I would argue that first person shooters are a dying genre on PCs through just looking from the competitive scene with gaming. While there is still may be a lasting Counter Strike scene, the focus has gone to the latest Halo or Call of Duty being played on a console for competitive FPSs.
It just happens to be that some game genres really don't work well on the console no matter how hard you try. Halo Wars was well done as a console designed RTS, but just doesn't have the same depth as its PC counterparts such as Starcraft or Warcraft due to the restrictions on the interface and controls.
Even if you do manage to design for the platform, a lot of games just don't work.
[1] http://www.vgchartz.com/game/35315/battlefield-3/ [2] http://www.vgchartz.com/game/40231/battlefield-3/ [3] http://www.vgchartz.com/game/40230/battlefield-3/
Microsoft tried cross platform halo but it didn't work that well: http://www.maximumpc.com/article/news/rumor_microsoft_killed...