Actually, there are 2 financial center in London. Canary Wharf and the City.

Canary Wharf does not enjoy the same weirdo status as the City and that does not seem to prevent bank like HSBC to be headquartered there. Also quite a lot of financial institutions bleed outside of the City, like in SouthBank.

The whole financial sector in the UK has a murky reputation, and there have been regular show on TV about how "the City" is a big money laundering machine. The City in this cased was used as a synonym for the whole sector in the UK rather than the physical location.