From what I have been reading about, the taking on of the husband’s last name is an extension of the wedding ring; both meant to signify that the woman is now her husband’s property (he “took her hand in marriage”), instead of her father’s. Women used to be kidnapped from their families for marriage. Rings were also worn by slaves in some ancient cultures, such as through the ears. This makes me think that the wedding band for husbands is a relatively new invention. And “patriarch” means male head of the family, which is the first line of institutional patriarchy. He owned women, their children, and slaves, and could also marry slaves.
Plus there’s the fact that the bride typically wears a white veil, symbolizing purity, just as if she were a gift being unwrapped. Parents, but especially fathers, love to get up in their daughter’s business about her sex life and claim they need to protect their “little girl.” It’s creepy, a violation of privacy, and incestuous-sounding.
When I was briefly married, I never changed my last name. I suspect this had something to do with the delay of the judge granting the divorce. That there needed to be a hearing and the scum didn’t show up because he was on military leave, is ridiculous.
Anyway, if couples are really bothered by this and don’t want to do either, they can hyphenate their names.
See also types of marriages in Biblical times. Might I add, the law allowing a rapist to marry the victim was alive and well in Morocco and was appealed only recently, on account of the suicides.