Not sure what more recent xboxes are like, but the original xbox was a cut down windows 2000 running on Intel and Nvidia. It was really close to commodity hardware and software.
In terms of hardware, they're running on AMD x86 CPUs. AFAIK there isn't anything special about them, other than having a wider memory bus (they use GDDR5 as opposed to DDR4).
But it had a really thin OS layer, and everybody had virtually exactly the same box, so you could micro-optimize to the exact architecture. IMO it does fit the concept of "domain specific stack", it's just that homogeneity is one of the important properties of the stack instead of unbridled performance.
