Original DOOM and DOOM II game engines were 2.5 D.
You could not look up and down or jump.
That is not to say that there was no z-axis in the maps themselves, but the game engine handled it in a simplified, flattened way.

Its been a really, really long time since Quake z-axis code was backported to create zDOOM.
You are a great example of people not realizing this.

Yes, the key aspects of 2.5D are that two things cannot be on top of each other, the world is entirely described by a 2D map, along with a height value for each floor and each object (these are separate). Jumping and looking are just not features included in the original game, but they are not defining properties of 2.5D, they could have been coded in and still be 2.5D

You can go up and down on lifts in doom, there is a height property that changes, and if you step off a ledge you fall in an arc to the ground. If an object is in front of the ledge somewhere at ground level you will collide with it even if you are higher, so eg if you are trying to jump across somewhere a mob far below can cut your jump short and you fall to the ground.