It seems to me that the birth certificate's purpose is to give a child a legal identity
A birth certificate is a copy of information in the register of births in the same way that a death certificate is a copy of information in the register of deaths. If you had read the thread you would realise that adoption does not change the information on the record of births, and that a short birth certificate does not give any information on parents.
just the fact that their traditional views on gender and sex are being challenged. It would be simpler to just list the parent who gave birth and the second parent, without mention of gender.
It's true that 'Mother' has different meanings to different people, but the sense used on the birth certificate just means the parent who gave birth. A birth certificate gives details about sex, not gender. Trying to impose gender on to a birth certificate is the epitomy of forcing parents into traditional gender roles, and that is why I object so strongly. McConnell clearly has a narrow view of what it means to be a man or a woman, and obviously is free to hold those beliefs. However a birth certificate does not and should not dictate gender or parental roles, it just records the facts of a birth.
I'm also confused as to why the birth certificate absolutely must list the biological mother, but the second parent apparently can be whatever. Is fatherhood somehow less valid or relevant than motherhood ?
Again, you are insisting that a birth certificate can give value or identity.
'Motherhood' is completely irrelevant to a birth certificate. The certificate is a record of birth, not a certificate of the mother's ongoing social relationship with the child. For many reasons some children do not have a relationship with their birth mother after birth.
The second parent listed on the birth certificate has a legal relationship with the child, but the nature of the mammalian reproductive process means that some males do not have a relationship with their offspring after conception, and some don't even have a relationship with the mother at conception.
Many people have parents listed on a birth certificate with whom they have no relationship, but the birth certificate doesn't change because it is an official document, not a scrapbooking memento.