Much like single-name celebrities Madonna, Prince, or Beyoncé, it’s infrequent that you see a member of the royal family attached to a last name. But rest assured, they do have them—and there's a complicated history attached.
Prior to 1917, royals didn’t have surnames; they simply went by the name of their house or dynasty (House of Tudor, House of York, etc.), which is inherited from the sovereign’s father.
In 1917, however, King George V made a major shakeup in royal moniker policy when he decided to change the house’s name from Saxe-Coburg-Gotha to Windsor (as in royal property Windsor Castle). Along with the name change, George V motioned that Windsor would also stand as the surname of all male descendants.
This carried on until Queen Elizabeth II took the throne. Several years into her reign, she and her husband, Prince Philip (whose last name is Mountbatten) decided that their particular branch of descendants would bear a modified surname reflecting both of their ancestries: Mountbatten-Windsor.
However, if the royal’s first name is preceded with Her or His Royal Highness, they don’t need to use a surname. Mountbatten-Windsor can be used on legal documents, but it’s not necessary.
And there’s more. Some royals choose to use their territorial designation as their last name. When Prince William and Prince Harry served in the military, they took the surname Wales after their father, the Prince of Wales. So technically Prince George and Princess Charlotte could go by George and Charlotte Cambridge—classy, no?
The thing is, there’s no absolute when it comes to surnames. Some royal descendants simply choose to use their father’s last name (Princess Anne’s children took the surname Phillips), while others strictly use Moutbatten-Windsor, or just Windsor.
No word yet on what last name royal baby no. 3 will prefer.