Do I need to keep Xbox 360 discs after I install them on Xbox Series X?

Yes, you must keep the disc, as it is then used as more of a license key, instead of actually loading in the game. They created this feature so people wouldn't be encouraged to just load in their friend's disc once and then have complete ownership, without paying anything. This proof-of-ownership system is found in every modern day console, such as the Switch, Xbox One, PS4 and PS5.