It mostly would.

Hungary renamed an iconic square in Budapest like 10 years ago. Most of the city still calls it Moscow Square.

if so wouldn’t that just end up with the USA following strange semantics other countries don’t?

When it comes to renaming external locations like this that are used by multiple countries, I would imagine it would be a case of how wide the rename is adopted. If I had to take a guess, renaming the Gulf of Mexico or the Atlantic Ocean in more US-centric names isn't as likely to catch on globally and will probably just result in confusion when people from different countries are referring to the same thing.

If America doesn't have to adopt the metric system they sure as shit don't have to name everything the same.

So, the way a geographical feature gets named, or renamed, is if the rest of the world recognises it as such.

Sometimes a name change of a feature, or the annexation of a region, gets internationally recognised not because everyone agrees to it, but everyone wants to stay on diplomatic good terms.

There's a similar situation going on with the Persian Gulf/Arabic Gulf.

The US has a law that forces other countries to abide by sanctions over trade. Netherlands and Taiwan are "colonized" by their tech industries that have a little of legacy IP from US companies. US companies can only license internationally with the stipulation that US rules over the licensee. 100% tariffs or military invasion promise for any country that disrespects America's rights to name stuff.

Everyone can call geographical features however they like in principle.

There's something similar here: https://en.wikipedia.org/wiki/Sea_of_Japan_naming_dispute

Countries can decide internally to call something whatever they want. One example is how not everyone in the world agrees on how many continents there are and uses some different names (like Eurasia). A more relevant case is how most of the world calls the ocean between Japan and Korea the Sea of Japan, whereas Koreans call it the East Sea.