Does a father have the same right to a child if it is born out of marriage, but his name is on birth certificate as a child born in the marriage?

"rights" the rights he has are those given to him by the court.
If his name is on the birth certificate he does not have to prove paterinty by a DNA test. But if you mean child custody, child support and the such, being marrried has no bearing on that what so ever