I've been to the City of London many times (though I worked outside it, in Mayfair).
It's really not "essentially sovereign" or "outside national law" as you might argue of an embassy or a military base or even, say, the Channel Islands. People and businesses there are subject to all the laws of the land. Financial firms there are regulated by the FCA, which is based outside the City and appointed by the Treasury, definitely not by the City of London Corporation.
Prestigious financial firms have offices in the City because everyone else does, not because there's some legal or regulatory benefit from being incorporated there - again, very different from those who choose other Crown jurisdictions like Jersey or IOM or the Caymans.
You're right, that was more of a knee-jerk reaction to a statement that it's like regular city governments in the US. The truth is more subtle: the UK benefits from The City's position as a world financial centre, and so parliament is incentivized to give The City a large degree of independence when it comes to setting financial regulation. By keeping them at arm's length they can have their cake and eat it too by setting strict tax rules for the entire country, but leaving loopholes to allow all the international money to flow in.
You can replace "The City" by "the financial sector" here. Once again, the historical curiosity that the City of London has its own local government has nothing to do with the UK's approach to financial regulation. Barclays, Citigroup, and HSBC - all headquartered 3 miles to the east in Canary Wharf, like the FCA itself, instead of in the City - get just the same regulatory benefits and disadvantages as Lloyds or NatWest.
It's really not "essentially sovereign" or "outside national law" as you might argue of an embassy or a military base or even, say, the Channel Islands. People and businesses there are subject to all the laws of the land. Financial firms there are regulated by the FCA, which is based outside the City and appointed by the Treasury, definitely not by the City of London Corporation.
Prestigious financial firms have offices in the City because everyone else does, not because there's some legal or regulatory benefit from being incorporated there - again, very different from those who choose other Crown jurisdictions like Jersey or IOM or the Caymans.