The idea of being named after your father (or your husband for that matter) is much older than surnames. Don't know exactly how old, because written history is only available for strong centralized states, but at least biblical times.
Dunno about other cultures, but in Poland it was very common until relatively recently to refer to wives/daughters by a suffixed first name (not surname) of the husband/father. If Mary Higgins married Jack Smith, from then on she'd be known as Jackowa, and their daughter Betty would be called Jackówna. This is distinct from naming children after their ancestors. The custom predates surnames by hundreds of years, but served essentially the same role of tying the women to their men.