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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.
Royalty does not use last names. Both princes belong to the House of Windsor. Their father Prince Charles is the Prince of Wales, which is the title of the heir to the throne. To distinguish which branch of the House of Windsor they belong to, they are called Prince William of Wales and Prince Harry of Wales, taking the name from their father's title. There is a difference between a name and a title.