We never even used surnames until relatively recently (early 19th century) – at least Ashkenazim didn’t. Indeed, if the woman was supporting the family (common among bnei Torah) a man was sometimes know as “[woman’s name]-man.” A woman’s Jewish name never changes (always her personal names and her patronymic). I don’t see how there can be a halachic issue here.
Practically, it is useful to conform to the local custom, especially in dealing with insurance companies and schools, who really like the name of the student to match the parents (or the name of the insured to match the person under whose name the policy is held). But that’s practical.