The British royals do not have surnames the way that most people do. Descendants of the Royal Family in the male line who do not have the style of Royal Highness or the title of Prince(ss) have the surname Windsor or Mountbatten-Windsor (for the descendants of Elizabeth II).

The birth registrations for those born as HRH and Prince(ss) have never included anything in the surname box, with the titles and given names and titular designation all given under the given names section.

I think on the registration of his daughter’s birth in California that they basically crammed “HRH The Duke of Sussex” into the various name fields.