AFAIK, the original XBOX was always based on a heavily stripped-down win2k kernel, and evolved very little. The "API" games used was a jump table to kernel functions, everything ran in kernel-space. I don't know much about later consoles, but I wouldn't be surprised if they used the same generic architecture.

https://xboxdevwiki.net/Kernel that site has quite a bit of info, with ongoing reverse-engineering and clean-room reimplementation projects (reactos, xquemu)