Most Jews, and all Ashkenazim, did not use inherited surnames until the 19th century. A man’s legal names was “Piloni ben Piloni” and a woman’s name was “Pilonis bas Pilonis”. While people often used additional names, they were not necessarily inherited. The goyim required us to use surnames in the 19th century to facilitate conscription, tax collection and assimilation. They initially forgot to require that they be inherited on the male line and it wasn’t uncommon for a man to adopt the wife’s family name (especially if he was moving in with her family and supported by them, as was common). Until recently, Dina Malchusa Dina required a woman to use her husband’s surname, reflecting various Christian legal traditions, but that is often not the case.
There is certainly no halachic requirement for a woman to adopt her husband’s surname, but it is convenient to do so.