It's pretty common in the United Kingdom for children to take on their father's last name. It's also not unheard of that children might also share their mother's last name, but as usual, the Royals do things a little differently.
The British royal family doesn't necessarily have a single last name. Technically, it's Windsor, but that's actually only a recently new development.
Before 1917, the royals actually went by the territory they ruled or the Royal House of which they were a member.
Queen Victoria's eldest son was actually known as Albert Edward Saxe-Coburg-Gotha. Imagine learning to spell that one as a child.
Given that the country was in the middle of World War I, King Edward's son, George V's German-sounding last name needed to be changed.
So, he decided to name his family after Windsor Castle. Now, any descendants of Queen Victoria are known as Windsors.
It gets even more confusing...
In 1960, Queen Elizabeth II and her husband, Prince Philip, decided to distinguish their descendants from the rest of the family by adding on Mountbatten to the Windsor surname. Her children and their children are Mountbatten-Windsors.
And just because they can, the royals sometimes adopt other names as they see fit. That's why Princes Willam and Harry wore tags that bore the surname Wales - an adoption of their father's designation as the Prince of Wales.
[h/t mental floss]