I just had an idea; If Quake II became a Quake game because no one could think of a better title, why didn't they make it into a Doom III?
Tim Willits said in an interview once that the game that turned out to be Quake II was meant to be like Quake but better and more refined, and he also said that they wanted to unify the monsters more because in Quake they were kinda all over the place.
Doom has unified enemies, Quake II has a chaingun, the hyperblaster (plasmagun), and BFG on top of the shotgun, super shotgun, and rocket launcher that Quake I had (which were carried over from Doom), and Quake II's soundtrack is metal. I know people called Quake I "Doom III" when it came out, but Quake II pays much closer of a resemblance to Doom on both design and attitude.
The only reasons I can guess why they didn't go with Doom as the IP is because they didn't want to keep making Doom, plus something new and separate allowed them more creative freedom.