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The British Royal Family changed their last name from Saxe-Coburg-Gotha to Windsor in 1917. So Her Majesty's last name is Windsor. Windsor. Some of her family are entitled to use the name Mountbatten Windsor which was authorised by an order in council in 1960.
How'd this come about? Well, it's simple. King George V, the Queen's father, chose Windsor as the royal surname in 1917, and the Queen's husband Prince Phillip's surname is Mountbatten. There ya go! In William's case, he has a title, so he can take the hyphenated last name or leave it, depending on the situation. Same goes for Princess Charlotte.