His surname is Mountbatten Windsor, he just doesn't use it because he is a Duke and that supercedes it. When Archie was born that was his announced name.
I think they announced recently that they were all going to use Sussex as a surname, in the same way he and William were known and William and Harry Wales at school. But it isn't his family name, that's M-W.
It's all weird but once you introduce titles names go weird.