> City of London is no weirder than NYC ...

really, not weird? compared to what? the Vatican?

It's arguably the financial (offshore) capital of the world and home to over eleven thousand people.

The City of London lives inside of the city named London but the two Londons have separate city halls and elect separate mayors, who collect separate taxes to fund separate police who enforce separate laws.

The Mayor of the City of London is called 'The Right Honourable Lord Mayor of London' to match his fancy outfit.
He gets to ride in a golden carriage and work in a Guildhall while the mayor of London has to wear a suit, ride a bike and work in an office building.

The City of London also has its own flag and its own crest and gets to act more like one of the countries in the UK than just an oddly located city.
The corporation that runs the city of London is older than the United Kingdom by several hundred years.

It was created by Romans in 43 CE as "Londonimium" with their temples, public baths, roads, bridges and a wall to defend their work. It's this wall which is why the current City of London exists - for though the Romans came and the Romans went and kingdoms rose and kingdoms fell, the wall endured protecting the city within. After the Romans William the Conqueror came to Great Britain to begin modern British history he found the City of London, with its sturdy walls more challenging to defeat than farmers on open fields. So he agreed to recognize the rights and privileges City of Londoners were used to in return for the them recognizing him as the new King. Though after the negotiation, William quickly built towers around the City of London which were just as much about protecting William from the locals within as defending against the Vikings from without.

This started a thousand-year long tradition whereby Monarchs always reconfirmed that 'yes' the City of London is a special, unique place best left to its own business, while simultaneously distrusting it. Many a monarch thought the City of London was too powerful and rich. And one even built a new Capital city nearby, named Westminster, to compete with the City of London and hopefully, suck power and wealth away from it. This was the start of the second London.

As the centuries passed, Westminster grew and merged with nearby towns eventually surrounding the walled-in, and still separate City of London. But, people began to call the whole urban collection 'London' and the name became official when Parliament joined towns together under a single municipal government with a mayor. But, the mayor of London still doesn't have power over the tiny City of London which has rules and traditions like nowhere else in the country and possibly the world.

For example, the ruling monarch doesn't just enter the City of London on a whim, but instead asks for permission from the Lord Mayor at a ceremony. While it's not required by law, the ceremony is, unusual to say the least.

The City of London also has a representative in Parliament, the "Remembrancer", whose job it is to protects the City's special rights. Because of this, laws passed by Parliament sometimes don't apply to the City of London: most notably voting reforms ...

https://www.youtube.com/watch?v=LrObZ_HZZUc