Mountbatten-Windsor, the surname of Prince Charles, is technically Prince Harry and Prince William's official last name. Though the two typically aren't referred to by their last name (we all know who they are once you add "prince" in front of their names), the brothers used their surname during their school years and time served in the military. While commoners traditionally take their father's surname, the royals are anything but a usual family.
When Prince George started school, he used the last name "Cambridge" (William and Kate are the Duke and Duchess of Cambridge) for all official documents. Additionally, the royal tot is not referred to as 'His Royal Highness' in the classroom. #royalproblems.
This means that it's probable Prince Harry and Meghan Markle's future babies (!!) are going to take a different last name as well—most likely "Sussex," Harry and Meghan's official royal titles.
While the House of Windsor has been the official name of the royal family since 1917, the royals' last names have changed over time. When Queen Elizabeth married Prince Philip Mountbatten in 1947, the couple compounded the two surnames and made "Mountbatten-Windsor" the official last name of the royal family after Prince Philip gave up his royal title as Prince of Greece and Denmark.
Other members of the royal family have also been bestowed alternative last names on their wedding day. Prince Andrew and Sarah “Fergie” Ferguson became the Duke and Duchess of York after their wedding day—a title the Queen gave them after her own father, before he became King George VI. Similarly, the Queen gave Prince Edward married Sophie Rhys-Jones the title of Earl and Countess of Wessex when they tied the knot in 1999.