The game was developed using the Unity engine specifically for Xbox One and PC. Because of the hardware architecture differences, a native version for the Xbox 360 does not exist. 🛑 The "RGH" Misconception
If you insist on experiencing Cuphead on a retro Microsoft console, get an (which can be found for under $100 used) or install Windows via Retroarch on Series S|X dev mode. The Xbox 360, despite its legendary status, must be retired for this particular title.
is not officially available on the Xbox 360 . It was released in 2017 for Xbox One and PC, which use newer hardware than the 360 [1, 3].