According to the legal text from Apple’s website, Apple makes Game Center nicknames (as well as “other unique identifiers”) available to app developers. Since these nicknames are unique! and tied to our Apple ID’s, I am concerned that this is a mechanism that app developers can use to indirectly identify indirectly our Apple ID’s and devices. And if you don’t think you have a Game Center login, you do, and you are signed into it. It is not possible to not have one, nor is it possible to delete it.