He is Prince Archie as his dad is a Prince
I don't think he is automatically a prince as great-grandchildren don't automatically get the title, or even to be called HRH. He'd have to wait until Charles became king, I think.
He can use Earl of Dunbarton from his father's title but he was announced as Master by his parents.
So I think like Princess Anne, Harry & Meghan have chosen not to have titles for their children. Such a sensible decision IMO - he can do as he wants in the future, unlike his father