It's not that different. There are 43 police forces in the country, with different specialities, and the City of London force leads (not very well) on financial crime. Businesses do have a vote in local elections, but that was standard across the UK until the Representation of the People Act 1969. The Corporation of London is basically just a local council like any other, albeit with a load of money and a load of meaningless ceremonies (both for historical reasons).
Well on the one hand you're saying it's not that strange, on the other there's the election process for who can become Lord Mayor of the City of London. ;)
In most parts of the country the "Worshipful Society of Apothecaries" plays very little part in local governance...
Here's another video from CGP Grey that explains it rather well:-
In theory the Lord Mayor is elected by 20,000 Liverymen, in practice s/he is chosen by Aldermen from amongst themselves.
27 London Boroughs have a mayor (City of Westminster a Lord mayor) chosen by councillors from amongst themselves. ~~~ CPG Grey is a bit of fun, not a documentary.
Also not true - if you are lawful resident in the UK you can vote in local elections, just not in the parlimentary elections(source: Polish immigrant who lived here for over a decade without a British citizenship, I have always voted in local elections).