That's somewhat different, though. The City of London is not London, the capital city of the UK. The latter is formally Greater London, consisting of 32 London Boroughs and the City of London. The two are different administrative entities entirely, with City of London e.g. having its own Lord Mayor separate from the Mayor of London, and with the ceremonial country of Greater London explicitly excluding City of London.

Nested administrative boundaries is a UK speciality (e.g. England, Wales and Scotland are countries within a country).

The 'City of London' is a quirk it is actually a very small borough within the city of Greater London, which has a population of something like 8 million and is very roughly up to zone 6 on a tube map. For the regional population I vaguely remember a statistic that there are 20 million people within 30 miles of trafalgar square, but I can't back this up right now.

That City (big C, the administrative area) is indeed tiny, population 8,000, and is distinct from the geographical city of London (small c) population 8,000,000.

And, notably, predates the English monarchy and technically exists somewhat separate from the UK...