Passports are basically letters from heads of state requesting safe access for their citizen who holds that letter. The oldest record of such being recorded in the Old Testament book of Nehemiah.

Nehemiah 2:7-10

And I said to the king, “If it pleases the king, let letters be given me to the governors of the province Beyond the River, that they may let me pass through until I come to Judah, and a letter to Asaph, the keeper of the king’s forest, that he may give me timber to make beams for the gates of the fortress of the temple, and for the wall of the city, and for the house that I shall occupy.” And the king granted me what I asked, for the good hand of my God was upon me. Then I came to the governors of the province Beyond the River and gave them the king’s letters. Now the king had sent with me officers of the army and horsemen. But when Sanballat the Horonite and Tobiah the Ammonite servant heard this, it displeased them greatly that someone had come to seek the welfare of the people of Israel.

The front cover of your British passport should include the following:

His Britannic Majesty’s Secretary of State requests and requires in the name of His Majesty all those whom it may concern to allow the bearer to pass freely without let or hindrance, and to afford the bearer such assistance and protection as may be necessary.

I believe other countries have a similar note.

Over time, they got more and more standardised into identity documents, but that front cover is basically the core part.

So, the King doesn’t need a letter from himself asking for safe passage.