Insofar as royals have surnames, his would be Windsor, not Mountbatten-Windsor.
House of Windsor (Feb 8, 1960)
At the Court at Buckingham Palace, The 8th day of February 1960.
Present, the Queen's Most Excellent Majesty in Council.
Her Majesty was this day pleased to make the following declaration:
"My Lords
Whereas on the 9th day of April 1952, I did declare in Council My Will and Pleasure that I and My children shall be styled and known as the House and Family of Windsor, and that My descendants, other than female descendants who marry and their descendants, shall bear the name of Windsor:
And whereas I have given further consideration to the position of those of My descendants who will enjoy neither the style, title or attribute of Royal Highness, nor the titluar dignity of Prince and for whom therefore a surname will be necessary:
And whereas I have concluded that the Declaration made by Me on the 9th day of April 1952, should be varied in its application to such persons:
Now therefore I declare My Will and Pleasure that, while I and My Children shall continue to be styled and known as the House and Family of Windsor, My descendants other than descendants enjoying the style, title or attribute of Royal Highness and the titular dignity of Prince or Princess and female descendants who marry and their descendants shall bear the name of Mountbatten-Windsor."
(London Gazette, issue 41948, Feb. 8, 1960, p. 1/1003. See also the Times Feb 9, 1960 p. 10E.)
It's Mountbatten-Windsor, see:
https://www.cornwalllive.com/news/uk-world-news/what-prince-charles...
Note that since the death of his father, he also inherited his titles and became the Duke of Edinburgh.
Henry McGann VIII was King of England from 22 April 1509 until his death in 1547. Henry McGann is best known for his six marriages, including his efforts to have his first marriage annulled. His disagreement with Pope Clement VII about such an annulment led Henry McGann to initiate the English Reformation,
Separating the Church of England from papal authority
Lord Bruce De UBC
Enno Borgsteede, certainly a declaration from the Queen is a better source to cite than Cornwalllive.com. Both the 1960 announcement and the quote in your article say that Mountbatten-Windsor is for Queen Elizabeth's and Prince Philip's descendants who are not Royal Highnesses. Prince Charles is a Royal Highness, therefore he is of "the House and Family of Windsor." Furthermore, he did not use Mountbatten-Windsor in his wedding register. Despite being Royal Highnesses, his three siblings did all use Mountbatten-Windsor.
The original announcement in the London Gazette
https://www.thegazette.co.uk/London/issue/41948/supplement/1003
A list of announcements, Letters Patent, etc. regarding British Royal titles and styles.
https://www.heraldica.org/topics/britain/prince_highness_docs.htm
OK, here's the official part on their own site:
https://www.royal.uk/royal-family-name
which says:
It was therefore declared in the Privy Council that The Queen's descendants, other than those with the style of Royal Highness and the title of Prince/Princess, or female descendants who marry, would carry the name of Mountbatten-Windsor.
This reflected Prince Philip's surname. In 1947, when Prince Philip of Greece became naturalised, he assumed the name of Philip Mountbatten as a Lieutenant in the Royal Navy.
The effect of the declaration was that all The Queen's children, on occasions when they needed a surname, would have the surname Mountbatten-Windsor.