Apple has quietly tweaked the Game Center terms of service to add in the rule that whenever you first invite a friend to join your friends list in the gaming social network app, your real first and last names (as entered in your iTunes account) are shown to them. You still use a nickname as a full ID (and Game Center will still show your nickname when you log in to browse achievements or see what people are playing), but when you first "friend" someone, they get to see your real name.
Apple has been able to get away with this one somehow -- the iTunes-based Ping social network also makes liberal use of the real name associated with an Apple account when adding friends, but for some reason nobody has made too much of a fuss about it. Consider Apple lucky -- when Blizzard Entertainment tried to require real names to be shared on their World of Warcraft forums, their community raised such a clamor that they had to back down and keep the forums anonymous.
But Apple hasn't had that problem, either because the audiences are different, or just because people don't care so much about anonymity among their Game Center or Ping friends. We'll have to see how the community responds to a change like this.