As you say, he's the actual son of Hamlet, but we're not given any information in the play about what family he belonged to. Royals usually didn't go by their surnames in medieval and early modern Europe. It was a real and deliberate insult to Lous XVI to call him "Citoyen Louis Capet" during his trial and execution, and the same was true of Charles Stuart, aka Charles I of England. They thought of themselves as belonging to royal houses more so than tight-knit nuclear families like everyone else.
I don't think this is what Shakespeare had in mind when he was writing Hamlet, after all, many of his characters go by what we would usually associate with first names only: Horatio or Polonius in Hamlet, for example. Plus, we never learn what King Lear's daughters' last names are either. It seems to be a detail that Shakespeare wasn't all that concerned with.
Thank you for the information.