1. Previously, royal children had no last names.

Why? Because that was customary until the 1900s.
Exception: After 1900, there was no requirement.
This is an intriguing custom that is currently obsolete. Royal offspring did not have last names of their own until the 1900s. Instead of using their last name, they were referred to by the House from whence they originated.
Royal Offspring Previously Had No Last Names © Tim Graham via Getty Photos
Following that, the royal family homeschooled their kids until Princess Diana enrolled the princes in private education. Currently, last names derived from the house are not required. For instance, Prince George is called George Cambridge rather than George Windsor when he is in school.