My impression is that more often than not throughout human history, at least in the West (including Russia), women kept at least some marker of their father's family or clan name, just as they kept some kind of indelible rights to a paternal dowry.
The more important extended family/clan was, and the more marriage was seen as a social contract between two families, the more likely that a woman's name could never be subsumed by her husband's. Marriage was a unity of two houses, not merely two individuals.
Even today, though, where none of this obtains, I think it's safe to say that most (or at least a very large plurality of) women on Earth retain some marker of their father's house in their full name, particularly since the Chinese do so. This is true in the majority of Orthodox lands, e.g. Russia (patronymic), Greece (keep paternal surname), Middle East (keep paternal surname), etc., and also many Catholic ones.