It doesn’t really function as its own state since it has no separate courts (it owns and partially runs the Old Bailey, but that’s for all of England, not just the City), no defense or foreign-policy capacity, and none of the other functions of a state. It’s a Liberty (an area with enhanced or different rights) rather than a state.

It’s the same for England, Wales, Scotland and Northern Ireland. Or for 50 states inside the US.