Hello there, TobyFromGB.
The following Knowledge Base article has some information on Signing into Game Center:
Sign in to your Game Center account
When you open Game Center, you can sign in to your Game Center account using your Apple ID. If you don't have an Apple ID or another Apple account, tap or click Create New Account to create an account to use with Game Center.
Note: Please note each Apple ID has its own associated scores, achievements, and friends. If you have multiple Apple IDs, you can verify which Apple ID is in use by tapping or clicking the Me icon.
You can change the account by tapping or clicking Account and then signing out of the current account and into the correct one.
You can also go to Settings>Game Center and sign out/sign in from there.
Thanks for reaching out to Apple Support Communities.