I don't want Google naming things, but I don't want a President naming them either. My preference is for things to be named by the people who live there/use the thing, but failing that at least I would want my mapping software to present the fact that there are multiple names/borders and let me pick which I want to view.
The president was given this power by the people. Crowd sourcing this is a particularly bad idea as countries already have working institutions that map and name things.
One alternative would be for Congress to create more barriers for name changes.
The president is taking this issue as a power grab. He does not have rights to name things. There is a geological naming board appointed by cabinet heads. The executive order instructs his newly appointed department heads to purge their geological naming boards of any members who may not agree to these names. This is not something that has happened before, and not by design.
Is "gets to rename any geographical feature in the general vicinity of the country on a whim" a usual power of the US President? I genuinely don't know the legalities around this, but given that, I would bet that most people who voted for Trump also didn't think they were giving him that power. (Whether they would have chosen to do so is a separate question.)
Do electors of any politician ever understand the full extent of their candidate's powers? They vote on people under the assumption that the person is the best choice (not necessarily a "good" choice) for the job given the system already in place.
Answering your question: yes, the executive branch can change geographical names in the US. Obama did that too with Mt Mckinley. This power extends solely to the names as accepted by the United States. If Mexico wants to change the gulf's name to "Gulf of the Aztecs" or "Gulf of qetwyetwdhxysysheussg", it can do that too.
SHOULD they do it? Now that's a different question.
