When you sign in with your Apple ID, you will be signed in to Game Center automatically.
• Game Center & Privacy - Apple Support
https://support.apple.com/en-us/HT210669
Curious how this could be set-up, unless an earlier owner briefly had use of the new mac.
If you took delivery from an authorized Apple reseller, and item may have been returned
then it could have been sold again and not erased. (Apple Retail by policy won't do that.)
Tried to delete the Game Center user account?
"Try deleting the account while in safe mode. If you're still unable to delete the account, try restarting
normally and trying once more, as sometimes after being in Safe mode; and then restarting normally
can resolve issues" ~ from apple host reply to ASC user question
• Use safe mode to isolate issues with your Mac - Apple Support
https://support.apple.com/HT201262
The other option, may be should you discover more odd effects in areas of your macOS; indicators
of a pre-owned Mac, arrange to return the unit to the reseller. (Erase and reinstall the macOS. ~ Or
see information details about using recovery About macOS Recovery.) Consider Contact Support