IMPORTANT NOTE: We have changed the Game Center terms and conditions to provide you notice that if you send a friend invitation, the full name associated with your Apple ID will be shared with the recipient. If you accept a friend invitation, the full name associated with your Apple ID will be shared with the sender.
We've tested things a bit and haven't yet found a full real name takeover in Game Center. Currently, real names don't seem to be appearing anywhere except on friend confirmation push alerts. Apple could potentially be updating the rest of the Game Center system currently, as it seems likely the other alerts will get similar treatment once all of these changes roll out.
We're going to be keeping a close eye on this over the next few days. It isn't clear if friends made previous to this change will have access to your real name, or if this is only going to be the case for future friends. Either way, it seems like a strange move on Apple's part considering that many gamers take anonymity seriously.