Yeah, I love those little quirky details about London... A few trivia I find funny:
Did you know that - technically - "London" does not exist? There is Greater London, which is not a city - it's either a county (when it includes the City of London) or a ceremonial county (when it doesn't). There is also the City of London, which is (as detailed in other comments) a completely separate entity and includes just a tiny bit of what is generally referred to as "London".
Another one: Quickly, what is the capital city of England? It doesn't have it, as England doesn't have its own parliament nor government, being ruled directly by those of the United Kingdom.
OK then - what is the capital city of the United Kingdom, especially if London does not exist? Well, it's the city where the Parliament and most of the government offices are located, and that is City of Westminster. While technically part of the Greater London county, it is a city in its own right.