Game Center sets the player name that you see in game. It is possible you were signed out of Game Center or someone else using your device signed in to Game Center.
1) Go to your device's settings menu and tap Game Center
2) Check that you are signed in to your Apple account and it is your ID that appears at the top
3) If this is not the case, sign out of that account and sign in to your account
4) To change your Game Center name, tap the name under Game Center Profile on this screen and change it there