Actually while Doom made the "3D" shooter popular, System Shock was the first fully 3D textured "shooter" that I can think of. And beat the pants off of Doom in all but framerate. Story and freedom of movement.
I think Descent was the first fully 3D "shooter" (more like a shooter/flight sim hybrid) that supported hardware 3D acceleration in a meaningful way.
However, you have to hand it to Doom for supporting the online mod community in a meaningful way. I remember playing the snot out of the various mods made for Doom 1 and 2 (basically the same engine).
As for storyline. Half-Life and System Shock 2 certainly raised the bar in the 3D accelerated era. And did it properly with a linear on rails movie type experience. Granted System Shock 2 had a limited amount of "freedom" but it was still a shooter on rails. I mean how can you tell a coherent and powerful story without being linear. Every shooter that has tried has failed horribly in trying to tell a compelling story.
And lest we forget. Epic's engine tech is also driven by one man only, Tim Sweeney. So in that they are very similar to ID. Both also having started with 2D scrollers.
Jill of the Jungle for Epic and Commander Keen for ID. But while ID made a gradual transition to full 3D (Keen -> Wolf3d -> Doom -> Quake) Epic made a much more ambitious transition going straight from 2D scrollers and pinball games to Unreal.
As far as engine tech. I'd have to say it's still dominated by ID and Epic at tier 1. And the likes of Valve and Crytek in Tier 2. Although it could be argued that Crytek is still a Tier 3 engine developer/licensor. I'm talking about the number of licenses sold of their respective engines. Monolith is still hanging around with their Lithtech engine, but I'm not sure I'd even consider them a tier 3 licensor of engines. And there's a few more engines hanging around in the Tier 3 licensing area from companies that I can't remember the names of.
Regards,
SB