Source ports add more 3d stuff to the game engine, like collision by cylinders with height (ie, a flying Cacodemon not would colide with you if it is flying above you), opnegl rendering, 3d camera, modern controls, slopes, etc. But at the core, keeps being a 2d game.
It's the difference between looking 3D and being 3D (well, being represented 3D in the game engine before rendering to your monitor).
Doom isn't 3D because the level maps can be squashed flat to a 2D grid. For every x,y position there was only one z for you and the mobs. There weren't any situations where a bridge or ramp took you/mobs over a lower spot you/mobs could also be at, rooms weren't stacked, etc.
That was fine, Doom pulled off its sleight-of-hand extremely well and the gameplay it offered was captivating!
You could argue there are no 3D graphics. Everything is transformed into a 2D screen representation.
Hence my squashing analogy, DOOM levels don't overlap and can be drawn ("squashed flat") on a 2D surface.
But you can't actually occupy a different height for a given location. This is NOT real 3D. That's why DOOM is sometimes described as a 2.5D game. Quake was the ID Software game that was 3D.
If you think DOOM is a 3D game, try creating a level that contains a player accessible location above/below another player accessible location, and see how the game renders it. Or, read the DOOM rendering engine links that are posted and pay close attention to the "not a true 3D engine" paragraph.
The format of the levels or the fact that there are no rooms over other rooms don't really factor into it.
I suppose at the extreme end, I'd have to explain why Wizardry on the Apple II is or isn't a 3D game, which would be more of a strained philosophical argument.
On other side, the original game engine, use 2d math with fixed point to rendering on a 386 CPU. Don't uses mvp matrix to render, like a real 3d game.
Take a look to : http://doom.wikia.com/wiki/Doom_rendering_engine