If it was to prevent accidental incest, they'd ask for both names, wouldn't they?
If they only ask for one, then the mother's would be safer to make sure it's not, you know, intentional incest. Lots of women have children from several different men, who grow up together. Besides, children get more DNA from the mother, so incest between half-siblings who share a mother would probably be worse.
Also, if a man doesn't know a child is his, then probably the child doesn't know, either. 
Anyway, that justification is utter nonsense, and I hope some gay men will start a petition to protest against being used as a lazy excuse for misogyny.
Don't disappoint me, guys! 