One issue that comes up frequently regarding custody issues from unwed parents - whether a father asking what his rights are, or a mother asking about her responsibilities to the father - is that, though the father hasn't legitimated the child, "his name is on the birth certificate." People often - mistakenly - believe that this means the father is the father for all purposes in the eyes of the law.
I understand this confusion - you've got one of the most official documents in a person's life, issued by the state, naming the father of the child. It seems like this would be the state's recognition that the father is in fact the father. But it's not.