Does/should the mother automatically have more ‘rights’ to the baby, because she carried him/her

So following on from another post, and many in the baby names page, I see many posts asking if the couple aren’t married, who’s surname should the baby have. Generally, the majority say ‘give the baby your last name if you’re not married’.

Why does the mother automatically assume that they have more rights to naming the child, than the father? Are mothers the more ‘important’ parent?

To be clear, I’m not talking about completely absent fathers, but even parents that aren’t together, why is the general consensus that baby should have mums last name?