They don't. Emergent did and all their assets were liquidated. Their original license most likely enabled them to keep using the foundation. Basically, the engine is built on Gamebyro but is heavily modified. Think of it like when Windows was still built on MSDOS. That's probably a good idea of how it is, but that may make things a bit more confusing.
Not quite. Gamebryo isn't actually an engine, it's a suite of third party middle-ware that must be assembled into an engine. This is why Bethesda has never referred to it as the Gamebryo engine, it's always been Morrowind, Oblivion, Fallout 3 engine. That fact is the majority of the engine's code, from the file structure to the tool set is all Bethesda. The only parts that ever belonged to Emergent Tech(original maker of Gamebryo) is the renderer and the file format. Everything else in the suite such as Havok, Speedtree and FaceGen can be licensed directly from their perspective owners.
Bethesda has rewritten from the ground up most of it's own code as well as replacing the middle-ware renderer with their own as well as ditching SpeedTree for there own 3D LOD tree system. However the engine's file structure remains the same and it must if we are to maintain uber mod-ability.
In summation, it's not Gamebryo that is at fault simply because Bethesda was using it. The engine is mostly Bethesda's doing and what they are doing with it isn't being done anywhere else. It's rough being a pioneer but Beth is getting better at it.