On my profile page it shows my birth name how can I take my birth name off?

If your game is connected to Facebook, that name comes from your Facebook account, and can only be seen by your FB friends. Other players will only see your island name and "not connected by Facebook". You cannot change that name. What some players do is have a separate Facebook account for gaming, but since your current game is already connected to Facebook, only customer service can attach it to another FB account.

If your game is not connected to Facebook, only you can see your name; everyone else just sees "not connected by Facebook."