Maybe that's true there, but my grandson has my son's last name and they were not married? Never did get married. Are you in the US?
(misread sorry)
Here in TN
if you are not married... the father has to be present and show id to sign the birth cert, he has 30 days to show id to sign or it has to changed through the court system
My son bio mom. has two more kids same father but the second one father was there but did not have id so he could not sign the birth cert
the first one has his name and the second one has her last name
if your married does not matter if the father is there or not, the birth cert gets father and mother on the birth cert