Family names weren't set things in the medieval period. I don't know much about the evolution of surnames -- especially in 14th century Italy -- but Aligheri is a patronymic, a name that indicates Dante is the son of Alighiero (like Stephenson used to mean that the person was the son of Stephen).

Leonardo's "last name" isn't da Vinci, either. He was born near the town of Vinci.