I've got a Bachelor of Computer Science degree, and I majored in Software Development. I have some understanding of how these things work.
This is a VERY simple 3D game engine compared to current standards, why would they go to the trouble of coding advanced collision detection algorithms to determine if an attack hits when they can just do something like "player A kicks, if player B is within X range in front of player A they get hit". In fact, if Capcom wanted they could almost keep their entire 2D gameplay code intact and just replace the graphical engine code, still using 2D hitboxes.
Yes, you
can completely disregard the third axis in a 3D game of this nature, simulating 2D hitboxes.
If Capcom is having trouble with collision detection, it's probably just because the Street Fighter development team don't have much experience with 3D fighting game engines. They've stuck to 2D for over 10 years, with the exception of Street Fighter EX games (which I'm sure they got a different team to develop).
By the way, even 2D PC games these days are rendered using Direct3D. Sprites are just 4 sided flat polygons with a sprite texture painted on them. In fact 3D graphics aren't even really 3D, they're 2D vector graphics which simulate 3D. Sprites are actually flat rectangular polygons with sprite textures, and even though it's a 3D game they still use 2D hitboxes.
EDIT: Actually, I just thought of the perfect example: Marvel vs Capcom 2 (as well as any other 2D fighting game with 3D backgrounds). They kept most of their 2D gameplay code and just replaced the graphical engine with a 3D one.