The birth certificate is not about genetics. It is about who has parental rights and responsibilities at the birth.
That is the woman who gave birth to the baby and forms a dyad with the baby. She is the legal mother regardless of whether the egg is hers or not.
Whoever she is married to will have shared parental rights and responsibilities for whatever children she bears whether there is a genetic relationship to the child or not.
If she is not married she can declare who the father is whether that is correct or not genetically and if he agrees to accepting paternal responsibility he can go with her to the registry office when she registers the baby.
As pp said, if there is any question of a difficult relationship or the mother doesn’t believe him to be a suitable father she doesn’t have to name him. Then he won’t have any rights or responsibilities towards the child. He can apply for parental responsibility through the courts, I think that will only be granted at that point if he is proven to be genetically related to the child.
Anyone interested can find more information on the UK government website.
Point is that a birth certificate legitimises (makes recognised by the state) the relationship between the child and his or her mother plus usually a father who can be presumed to have provided the sperm but not necessarily and it doesn’t matter much to the state that he did or not. It just means that that man or nowadays other woman will assist the mother in providing for, caring for, disciplining etc that child so that the state doesn’t have to.
Pregnancy and birth are the key elements of biology in the birth certificate (clue is in the name?) genetics are not important except in proving paternity where that is in dispute and one or other party wants it proven.