They have the ability to change name services in some database somewhere. But it's not actually an official name change.
So all this hand wringing about official names or whatever ignores the reality of the situation. This is just another temper tantrum by Trump that we reasonable people can ignore.
The easiest way to do that is to just say Gulf of Mexico.
Is it Congress who officially holds that power, then? And are you saying that if Congressional Republicans passed a bill renaming whatever, and Trump signed it, you'd then acknowlege it? Given how vanishingly little they accomplish in Congress already, I'm not sure we'd be better off distracting them from their semiannual debt ceiling/budget hostage situation in order to debate these inconsequential things.
This is a pretty mushy area, philosophically. Is a name what you think something is called (I think our planet is called “Bloop”, but nobody cares) what a lot of people think something is called (“Indian” in reference to indigenous Americans is a famous example of that going poorly, as are many racist epithets), or what an official body says it should be called (“Palestine”, “New Zealand”, and “Taiwan” have entered the chat)?
I don’t think popular usage is any more “truly authoritative” for a name than other sources.