No, it isn't. It's a strangely-constituted local authority. Combined with the "Guilds", which are effectively trade associations that hold dinners and occasionally public events in historical dress. It's a very
cliquey environment, but that doesn't make all the conspiracy theories true. When people say "the city", they often mean "the UK financial industry", in the way that Americans say "Detroit" meaning "the US car industry".
What the "city" may well get is favourable interpretation of complex ambiguous tax arrangements. That's the argument against e.g. Vodafone. What the "city" is afraid of is changes in EU law that seem highly technical but have the effect of damaging its business. The Tobin tax is one example.
(London in general benefits from the stupid way property tax is done in the UK with finite "bands", so a family house and a multistorey mansion pay the same annual tax, but that's a separate issue.)