Is this breaking the 20-year rule? checks Holy crap no it isn't. Doom came out in 1993, and is thus 22 years old this year. Wolf3D is a year older. I feel old.
Most of the notable games that Moby Games lists as being first-person shooters from 1992 are not what "FPS" means as a genre; I see a few flight sim combat games (F-15 Strike Eagle 3, Wing Commander 2, After Burner).
There is, however, 1992's Ultima Underworld, which was a pretty typical Looking Glass title in that it ran great on the average computer of three years after its release date. It also tends to not end up on "first person shooter" lists; folks tend want to put it in the 'dungeon crawl' category. On the other hand it also beat Wolf3D to market by about two months.
Meanwhile, "FPS on rails" had been a thriving genre in the arcade in the person of cabinets with big plastic guns mounted on them. I dunno if Taito's Operation Wolf (1987) was the first one, but it was a pretty notable one. (The first game to mount a plastic gun on the cabinet was Exidy's Crossbow in 1983, but you took the role of a never-endangered sniper picking off enemies attacking an adventuring party.)
And a very notable arcade game that felt pretty much like a modern FPS except for having your character on-screen was Atari's Xybots from 1997). Big mazes to wander around in, persistent enemies, side-strafing, secret doors, it's pretty much all there in a slightly more locked-down perspective.
And then there is Faceball 2000, a first-person free-roaming maze shooter from 1991 on the Gameboy. With a happy face theme. Which supported deathmatch if you had a link cable. It was a port of Midi Maze for the Atari ST from 1987. Which also supported deathmatch, via the ST's built-in MIDI port, hence the name.
After Doom, there was an explosion of FPS games. Or as we called them back in the 90s, "Doom clones". Wolf3D* and Doom took a bunch of ideas that had been floating around the video game world for a while and added quick and dirty 3D texture mapping, which turned out to be the thing that really stepped it up another level. They weren't the first, but they chose to make a lot of compromises on the 3D in the name of making it playable at a decent frame rate on an average computer, unlike Ultima Underworld's more sophisticated display.
Before Wolf3D, scaling and rotating bitmaps flat on was pretty much the domain of arcade games with special custom display hardware - Sega's After Burner is a pretty good example - and doing any kind of perspective transformation, even the quick and dirty work of Wolf3D or Doom, was impossible. MS-DOS machines of the day were starting to have just barely enough power to draw a texture-mapped 3D scene if you used a lot of shortcuts. Which Wolf3D did.