Wolfenstein 3d was the first major "FPS" game, released as shareware to begin with via BBS systems. It defined the genere as we currently know it. Doom was a successor to Wolfenstein.
However, as many super-long term game nerds can tell you there *WERE* games with a similar playstyle released before that, albiet none as polished or done to quite the same standards of "3d". Mostly failed attempts to create things similar to the "gun controller" games in actual arcades where you could control the movement.
While there might be things older there was a hacked together game kicked around BBS systems called "Assault On Smurf Village" which was a spiritual sequel of sorts to the "Castle Smurfenstein" mod/recreation of the original Silas Warner "Castle Wolfenstein". "Assault On Smurf Village" being very crude as befits this kind of "basement development for lulz" project but included a playstyle (gun in the middle, free roam) similar to Wolfenstein 3d, albiet it moved more like an old "Bard's Tale" game with no real aiming or smooth gliding movement. Back on things like Fidonet there was some discussion as to whether this was the inspiration for Wolfenstein 3d, though no definitive conclusion was ever arrived at to my knowledge. It does make a degree of sense then when you look at the liscence they used and what spawned from it.
You pretty much need to have had a C-64 though to have any idea about the original pre-FPS "Wolfenstein" game though, never mind any of the other stuff.