Can a married man give his last name to child born outside the marriage? I yes! What is the outcome of this action?

Can a married man give his last name to child born outside the marriage?. I yes! what is the outcome of this action?
What is the legal status of the child?If a man gives his last name on the child's birth certificate he will be the legal father and that means he has right for parenting time and obligation to pay child support.