Example Linux or Windows as the software is called thank you
The Xbox One has multiple operating systems. One of them is the normal Xbox operating system on which the games run, and there's also an adapted Windows 10.
The kernel is that of Windows 10, the interface was made suitable for the Xbox ONE u. SEries X
With the 360 it looks different because there's no Windows here either but an extra for the Xbox 360 developed on Directx based system is also parts of the windows source code but is just Power PC architecture from the processor there's no windows.
the classic first xbox had no operating system but only a small administration program the kernel was in every game and addressed the hardware directly, so it runs even when the hard drive is removed or you can just put a game on the hdd and boot it who is concerned with the box (modchip must be installed or softmod)