The Official Website of the British Monarchy
says:
Before 1917, members of the British Royal Family had no surname, but only the name of the house or dynasty to which they belonged.
Personally, I think the Official Website, and possibly the Royal Family itself, is reading to much into the proclamation of George the V by which name of the "Royal House and Family" was changed to Windsor.
The royals themselves are pretty sloppy about names. When Princess Anne married the first time, her surname appeared in the register as "Mountbatten-Windsor," which seems to be just a whim.
If we take the proclamation of George V as conferring a surname on the royal family, it would still only apply to the male-line descendants of Queen Victoria, and Princess Anne is not a male-line descendant of Queen Victoria. The announcement in 1960 that Queen Elizabeth II's descendants would bear the surname Mountbatten-Windsor specifically excepted those of her descendants with the style HRH. So before Princess Anne married, either she had no surname or her surname was Mountbatten. Likewise, Prince Charles, his brothers and their children either have no surname or their surname is Mountbatten. However, the members of the Royal Family, when they choose to use a surname, seem to use whatever surname they feel like using at the moment.
Mountbatten itself only became Prince Philip's surname in 1947 when he adopted that surname on renouncing his Greek and Danish titles. He was born a Prince of Greece and Denmark, and is a member of the House of Schleswig-Holstein-Sonderburg-Glücksburg. He chose to take the name Mountbatten -- the surname adopted by his maternal grandfather, the former Prince of Battenberg, when he (the grandfather) renounced his German titles during World War I.