Also the traditional purpose of an engagement seems to have shifted!
When we were engaged, 24 years ago, we encountered people who were really surprised - to the point of disbelief - when they learned that our being engaged actually meant that we were committed to marriage (ASAP after graduating). To them, you were either engaged (i.e. had been dating for a while and had no immediate plans to break up) or you were engaged to be married. We, in turn, were surprised that there was such a thing as being engaged without the clear intention/commitment to marrying. It would be pretty much like telling people you're pregnant and having a baby - you expect the single word to naturally convey the whole concept....
Tradition has always dictated that children take their mother's name; it's just that tradition has also dictated that couples marry before having children and also that wives take their husbands' surnames. If a traditionally-married woman who has taken her husband's surname subsequently has a baby, the baby is named after their mum - it just so happens that their dad also has the same surname. He was the source of their mother's current surname, but she is the source of theirs.
Where parents are not married, the law gives fathers no naming rights at all. In accordance with the law, registry offices will allow any of the following to legally register (and thus officially name) a baby:
A. The mother on her own;
B. The mother's husband on his own (whether or not he is the baby's father and/or shares the surname he legally declares to be given to the baby);
C. The mother accompanied by her husband - with both considered to have equal agency;
D. The mother accompanied by her partner/boyfriend - with her considered to have sole agency but with him allowed to be present and agree on name/registration details as a courtesy extended to him by the mother, who is always in the 'driving seat' with all decisions and has sole right to rule on all names/birth details, should her partner wish (unsuccessfully) to state/decide anything differently.
Essentially, the law considers a child to automatically be his/her mother's child (as biology obviously dictates) if she is unmarried; and if she is married, the law automatically extends this right equally to her husband.