Thank you Indie for the Na reference! A lot of societies past and present have functioned without the consept of paternity, often somewhat hazy on the mechanics of reproduction to begin with. Not long ago, men in Western societies were thought to place a fully-formed proto-human baby with their seed into the woman's fertile ground. Thus it was believed that the mother had no biological link to the child and she was just the incubation environment for the man's offspring.
I would think that in modern societies the number one problem with doing away with paternity would be genetic conditions that pass from parent to child. Thus I can see why it would be still necessary or beneficial to hold on to the idea of biological parenthood, but many lesbian and straight couples who have AId have managed to do without knowledge of the donor's identity, so I think it's more a matter of personal taste.
Me: bi female in my twenties