Duchess or Princess? Here is an interesting article on the subject, following the filing of the new baby's birth certificate: https://socialreader.com/me/content/7nFLO
"Can We Still Call Her Kate Middleton?"
The surname of the descendants of Queen Elizabeth II who bear royal styles and titles is Windsor (under the terms of the letterws patent of 1960). The name Mountbatten-Windsor only comes into play for the descendants who are not HRH or Prince or Princess.
The surname of Prince William is Windsor, and his wife and son are also Windsor.
It is interesting that when Prince was officially registered acouple of weeks ago no surname was actually given on the birth certificate.
I have found the following on "The official website of the British Monarchy" concerning the Royal Family name:
" ... all The Queen's children, on occasions when they needed a surname, would have the surname Mountbatten-Windsor."
"For the most part, members of the Royal Family who are entitled to the style and dignity of HRH Prince or Princess do not need a surname, but if at any time any of them do need a surname (such as upon marriage), that surname is Mountbatten-Windsor."
"Unless The Prince of Wales chooses to alter the present decisions when he becomes king, he will continue to be of the House of Windsor and his grandchildren will use the surname Mountbatten-Windsor."
I think the problem is that we are so used to filling boxes with first name, surname etc that many of us find it difficult to understand that some (very few) people do not have, or need, surnames!! :)