What personal identification documents do members of the Royal Family have? Does the card allow all members of the Royal Family to travel any place with just that card?
The Royal website partially covers this (emphasis mine)
As a British passport is issued in the name of Her Majesty, it is unnecessary for The Queen to possess one. All other members of the Royal Family, including The Duke of Edinburgh and The Prince of Wales, have passports.
Queen Elizabeth has visited more than 100 countries during her reign, but she has not needed to provide any proof of identity once.
This is a privilege for the reigning monarch only.
So it's safe to say that all other members of the family need a normal ID and/or license that any other UK citizen would need.