Isn't fleet more of a generic term for the naval resources of one particular government in a given area? Like the Pacific Fleet?
Currently, yes, although it's really only the US and Russia that separate their navies into fleets nowadays; they're the only people who have permanent large deployments in geographically dispersed areas, and even then, in Russia's case, it's entirely because they've got widely separated sea boarders. Hence things like the Black Sea Fleet (that, AFAIK, is actually a fleet in the older sense of the term). The US uses fleets as administrative units, rather than combat formations.
Traditionally, up until WW2, fleets were distinct combat formations. So, for example, Britain had the Home Fleet in the British isles, but also had the Channel Squadron and the Fisheries Protection Squadron and a bunch of other smaller units that weren't actually part of Home Fleet. Likewise, the Mediterranean Fleet operated in the Med (surpise!), but there were a bunch of other, smaller units not actually part of that fleet and the Eastern Fleet didn't include the Hong Kong Squadron, AFAIK. The US, of course, had the Pacific Fleet and the various numbered fleets, WW1 Germany had the Hochseeflotte, Russia had the Black Sea/Baltic/Far East fleets and so on, all of which were actual combat formations, not simply administrative. That sort of thing has vanished because there's basically nobody to fight a major fleet battle
with anymore. The US could take every other navy in the world in a stand up fight, and all of the second tier navies belong to US allies as well.
Additionally, the US is the only country with enough ships to make retaining that level of organisation worthwhile: Russia, as I said, does it for reasons of geography. Everybody else doesn't have fleets, or, more accurately, just has a single fleet that encompasses their entire navy, from which ships are detached to task forces or independent deployments.