>>Queen Elizabeth II doesn’t even have a driver’s license. As Queen, she doesn’t need one
Didn't know that either.
It's also true that she cannot be prosecuted for any crime except that of treason against the British people, but that's contestable. Since crimes are prosecuted in her name.
She perhaps didn't need one because other countries didn't insist on it when traveling, not because it is issued in her name.
Also the office of Her Majesty is distinct(and all other titles) from the person the queen herself. Monarchs are quite used to that idea, and that is how they award themselves other titles for example.
Same reason why royal name is assumed on ascension and does not have be birth name or her actual birthday is not when royal birthday is celebrated and myriad other things like that .