The insane part is that Amerigo Vespucci (the guy "America" is named for) never set foot in any territory that later became part of the USA. He "discovered" the coasts of Venezuela and Brazil, mostly. So, although a disputed account has him actually reaching the Gulf of Mexico, he never went anywhere near Lake Ontario.
And then with both of these explorers (Columbus and Amerigo), you have a weird stolen valor effect with the "we discovered it" idea. Most of those claiming the heritage as being "their people" are not from Spain, Portugal, or what later became Italy (Columbus was from Genoa financed by Spain, and Amerigo was from Florence, financed by Portugal).
So... why the word they're using even sparks nationalism isn't particularly clear.
But it's best not to even try to understand uber-nationalist Trump fanboys. They're far easier to manipulate than they are to understand.
And yeah, even Trump's ancestry in particular has no association with the countries that funded it, or the Pre-Italian explorers who made the trip. His ancestry looks a lot like mine (Saxon in my case... Germany didn't exist yet, but the Trump's were from the palantate, further south, Germany did exist (barely) by the time they moved accross the pond). His actual German and Scottish ancestors didn't "discover" "America" until the 19th century. So... not sure why he's trying to associate himself with explorers from areas that later became Italy that were working for countries he has also never had any association with.
So... perhaps we should deport the "Merica!" nationalists back to Venezuela (or Spain, Portugal, Florence... wherever is fine), since that's where they apparently think they came from? I guess that's one takeaway. The name of the actual country is The United States. America refers to the whole continent, but it's a bit less nominally appropriate for the land this country actually occupies than it is for other places.
If we wanted to rename Lake Ontario for the people who "discovered" it with the usual Eurocentric bias, that would be Samuel de Champlain and Etienne Brule in 1615. So... French people.