More than 50% of Americans think the woman should be legally required to take her husband's name in heterosexual marriages. The reason typically given is that having the same name increases a sense of family identity.
Making it into a legal requirement would be bizarre, but I agree such identity is important and sharing a name helps in its creation. Even on its own, marriage is, among other things, a way of tying yourself to the mast: deliberately making public declarations, taking vows, and arduously organizing an unnecessarily expensive party, all in order to increase your investment in each other, and make it more difficult to end the relationship during future difficult phases. Having the same name is one more way of making public and concrete your intention to stay together for the long haul.
But why should that mean that the woman takes the man's name in heterosexual marriages? Why should the man not take the woman's name or, as my fiancée and I have chosen to do, both choose a new name? (We've gone with "MacAskill", her maternal grandmother's maiden name. When I tell people I'm changing my name, I've met raised eyebrows, confusion, or aggressive questioning. No one's batted an eyelid when she's told others the same.)