Is the Game Center account on iPhone linked to your iTunes account or is it linked to something else?
Yes, it is linked to your iTunes account, as when you open Game Center on your iOS device for the first time, it asks you to log into an iTunes account. Facebook elements (referring to the 'like' function) in the app are linked to your Facebook account, separately% (if you have the Facebook app installed).
Because of this system (using iTunes accounts for login), it makes it possible for game saves/highscores to be synced^ between devices (even though iCloud is mainly responsible for it).
^ Not all games support this.
% I doubt your Facebook ever gets linked to your iTunes account (or vice versa).