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I discovered that when you create your Game Center account, it will use the name on the credit card associated with your Apple ID.
Changing the name in Game Center application does not update your visible name for the other users. What sequence of steps is needed to get my new name displayed on devices that had friended me before I changed my display name?
Have you tried contacting your credit card provider?
Seriously, Apple's intention is that iOS Game Center friends are meant for "real" friends—rather than "Facebook friends"—so exposing your real name is not a threat. If you look at the issue this way, it feels more of a security enhancement than a threat. You can be sure the friend you are friending is actually the person you believe them to be.
Moreover, if you are underaged and can't attach your own credit card with your Apple ID, it is, in contrast, more secure to have the user's (kiddie's) name obfuscated.
Finally some legalese from the Game Center TOS:
The full name associated with your Apple ID will also be shared with those users to whom you send, or from whom you accept, a friend request, such as in the Friends list, the Friends leaderboards, and the pending friends requests lists; and we may recommend games that you have played to your friends. If you prefer not to share your full name or profile with friends, you should remove that friend by tapping the “Unfriend” button at the bottom of that friend’s screen in Game Center.
For each friend's account you need to force an update, you should unfriend them and then establish the friend relationship again.
