Because they're the only country with America in their name whereas United States is shared by Mexico
Also in English there's a distinction between North and South America, with the supercontinent being referred to as "The Americas", so America isn't really ambiguous, they're also geologically and environmentally distinct enough that "The Americas" isn't used so much and "New World" is often more relevant to include Australia as another somewhat culturally similar sparsely populated former colonial area.
In Portuguese (as spoken in Portugal, Brazil, USA, Japan, Ghana, wherever) they're an americano/a but in English (as spoken in USA, UK, Brazil, Portugal, Nepal, wherever) they're South American but not American because it's a linguistic difference rather than a geographical/cultural one
Same reason why people from the United Kingdom are called ‘British’, despite Northern Ireland not being in Britain.
There just aren’t better proper adjectives for these 2 countries.
While you can say ‘North American’ to mean anyone from North America and not specifically the US, I’m not sure there’s a fitting word that refers to anyone from North or South America. Although, at that point, the group of people you are talking about is perhaps too broad to be useful in most cases.
Because there's always got to be some young "boo America" edgelord contrarian that thinks they're really doing something by saying it. But really they are just an idiot in a long line of idiots who isn't saying anything at all.
US Americans are seen by basically everyone as the most obnoxious tourists on this planet. Their self importance and undeserved national pride is second to none and it shows in every thread like this. Its just fun to see them get defensive when anyone points it out.
This is literally what that sign is about and its odd that you are confused that people would point this out.