If males get to be able to claim a woman's child just cause he dumped seed and demands to own the child, a woman's female partner should definitely be able to put her name on the birth certificate, seeing she unlike a male will actually be beneficial in the child's life.
Discussion
I lean to birth certificates are for the child to have a legal relationship and be able to access medical history. This is also a problem for persons who are adopted.... if birth certificates are ownership of children, there can be another field form of parental responsibility to have nonbiological adults who have a relationship.