I often see people, especially on YouTube, refer to the engine used in Wolfenstein 3D as "idTech 0", and likewise Doom is said to be "idTech 1". Popular convention so popular even Wikipedia uses it to some degree or another.
The engines used in Wolf3D and Doom are completely unlike each other, and in turn completely unlike Quake's.
Now, Half-Life was made based on Quake. This is common knowledge — they took QuakeWorld and bits of Quake II and that shortly after became known as GoldSrc. Then they developed the Source engine from that, and Source 2. And even without code access, just from watching others play, you can recognize that there's very specific things common to all of them, from bunny-hopping and other such typical schmovement, to famously the particular pattern in flickering lights.
There are many things in the Quake code that you can directly map to things in Source 2, via Source and GoldSrc.
Having seen the code for Wolfenstein 3D and Doom (and modded the former), I can honestly say that they are basically nothing alike. They are as different from each other as Quake is similar to Half-Life.
You'd have an easier time convincing me that Catacomb 3D and Wolfenstein 3D share a common engine, one being a (vast) improvement on the other but still sharing a ton of code.
idTech 4, the engine for Doom 3, was apparently actually called that when it was made. And it was id's fourth 3D game engine. It makes some sense to retroactively call the other four the same thing. Quake 3 Arena's engine becomes idTech 3, Quake's becomes idTech 2, Doom's becomes idTech 1, and then you run out and have to get cute with Wolfenstein 3D's engine becoming idTech 0.
But it's easier to trace a lineage from idTech 4 to 8, than it is to go back. Because from what I'm seeing Quake, Doom, and Wolfenstein have next to nothing in common besides being first person shooter games made by id software.
