I will tell you, I was in Michigan when I had my first baby, and there is a statement on the birth certificate application that says something like this in bold "Michigan is law is very clear: If you were married at the time of conception you MUST name your husband as the father of the child. If you were married at the time of the birth of the child you MUST name your husband as the father of the child." They even give you a flyer that explains the law, and how to change the name later if the father is actually someone else. In Mass, I am pretty sure that no matter who you name on the birth certificate your husband is presumed to be the father.
And, in Mass, if the husband or wife do not agree to start a paternity suit, the BF cannot file unless he has a significant connection to the child (which can be impossible in some cases as you could imagine). In Michigan it is even more strict, where only a wife or husband can even bring a paternity suit. BF cannot even file. If he does, it just gets thrown out. It has led to crazy (and sad) cases in the state. Even fathers who cared for their children, even one that I read who was the primary custodian for years, were not considered the legal father since the woman was married at the time of birth (and she decided she didn't want him involved anymore, and her husband did not contest for some reason). Even if there is DNA. Doesn't matter there. The supreme court of Michigan ruled that they had to uphold the law (this was in the case where the guy had informal custody for 3 years), but called for it to be changed since it led to such unfairness. So, the point is: the law across states varies a lot, especially in unmarried cases. Even more when the woman is married at the time of birth or conception. I thought about this a little bit, and I believe the original intent of these laws were to protect marriage and did not have much sympathy for men who were committing adultery with another man's wife. It was also to protect the husband, I think, so they could pass a child that resulted from an adulterous relationship off as their own, if they wanted.
In France, a woman can choose whether or not she wants the father to be involved as long as it is prior to the birth. She does not have to name a father at all. I know this because my husband's stepsister did this, and is the only legal parent of her child. I am pretty sure that the BF has no recourse if she decides this.
But I agree that as far as name goes, you can name a child anything (I am sure you heard of the child named "Tula does the hula in Hawaii" or something like that-this was changed by a judge to a name that the child preferred to go by). Personally, I don't even believe that a child should necessarily bear the name of the father. Think about that hard because it will be difficult to change later.