Technically speaking, Batocera runs on x86_64. The Xbox is x86_64, so there’s no platform limitation or anything. The real limitation is Microsoft not allowing custom firmwares running on the Xbox, as well as probably needing unique drivers for video output/controller input/bluetooth that are not compatible with the Linux kernel.
It’s a pipe-dream.
But, if you could run say VirtualBox on the Xbox, then you could probably run a Batocera image in that. I dunno how well its performance would be if you did that, though.