In doom and the at the time doom clones you couldn’t look up and down. The world looked 3d but technically wasn’t. They couldn’t even have rooms above each other because of this. The games were at a technical level top down shooters but viewed from the first person perspective of your character with graphical renderings likely using raycasting to give the illusion of a 3d space around you.

I’m not sure I’m following. I’ve never known Quake or Doom to have a camera independent of your player character’s perspective (at least without console commands or demo tools)

That’s exactly the point. Your perspective isn’t independent of your plane of movement.