This is the 21st Century. We should be away from the times where people who hold power are held to a different standard. Ultimately, she should have to follow the laws like everyone else. Even though she is queen, she is ultimately just a person. It would set a good example for everyone else to have her follow the rules and get a passport.
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. Using this logic—that the message inside a British passport makes it redundant for the Queen to carry one—shouldn’t John Kerry be able to go without, too? Here’s what American passports say: The Secretary of State of the United States of America hereby requests all whom it may concern to permit the citizen/national of the United States named herein to pass without delay or hindrance and in case of need to give all lawful aid and protection. Probably not. After all, the secretary of state is an employee of the U.S. government. let’s just imagine a customs officer asking Kerry what the purpose of his visit is, and Kerry saying that it’s to negotiate a nuclear deal with Iran.
Passports are a major inconvenience for most people, and attaining one takes a lot of time. While it may seem unfair for one person to be able to circumvent the process, Queen Elizabeth is the most recognized woman in Britain and British passports are issued under her name. It stands to reason that she would not be required to obtain one.
Of course the Queen doesn't need a passport! She is royalty, and I'm willing to bet that she travels with security teams and is always well known wherever she goes. The point of a passport is to have identification wherever you go, and I don't think she really needs that.