OK, seems to be easy enough... after logging in with you Apple ID you still have to configure your Game Center account So everything was OK, just need to give location, your desired user name and so on.
This happened to me this evening right after trying to invite a friend to play against me. I cannot figure out how to have it send me my user name or password, but it keeps insisting on trying to have me create a new Game Center account. Can somebody help?
I figured out the problem for me. I had two apple.com accounts. Since Apple doesn't give you a way to cancel an account, I had to login to Game Center with the one account and change my nickname to something bogus. Then I could log out and log in with my other apple.com account. Make sense?