I just don't understand how the resulting child is considered of the woman to whom the embryo is given. Wasn't the embryo inseminated already? and isn't the womb of the mother just being used? that would seem to me that she is bearing someone else' child. Maybe i'm not understanding something here.
Father, not being American, I'm only guessing, but there are probably statutes which create certain "presumptions" of parenthood in law.
For instance, the legal father of a child is presumed to be the husband of the natural mother unless there is evidence to rebut the presumption.
In New South Wales, I think it used to be the case that the "surrogate" mother was presumed to be the legal mother of the child, but that probably changed with the recent amendments to the surrogacy laws. I know that in times past, it was necessary for the genetic parents (the sperm and egg donors) to actually adopt their own child once it was born from the surrogate.