Board Thread:Support Requests - Getting Technical/@comment-46477043-20200729044328/@comment-9605025-20200729045109

Most mobile games have a login/logout separate from your Apple ID. Your Apple ID should be whichever Apple Inc. account you want the device associated with; not the account you want to use for a specific game.

Based on your description, I am guessing the full name of the game is "The Battle of Polytopia". I am not familiar with this game so I don't know for sure if it offers the ability to separate the game account login from the Apple ID login. If it does, I don't know how to access it. However, the people on The Battle of Polytopia Wiki might since they play the game like you and your son do. I suggest you ask there. If they can't help you, then you will need to contact the game developers (Mojang AB) or Apple.