Nowadays there is a separate space for the child's surname, but this wasn't the case back then. The child would therefore take the surname of the father (if named) or, if no father is named (as in this case), of the mother. The mother's married name was her legal name at that time and therefore the child's name too, however I wouldn't assume for a moment that means the father would be her husband. Given that he isn't named, I would strongly suspect that he isn't.
Ermy