Peru is American. South American, but of the Americas nonetheless.

The US gets the word applied to itself because the name of the continent is actually included in the name of the country, hence the "United States of America".. which lends itself to the semantics of it all getting a bit sloppy.

But.. Argentinians, Brazilians, Canadians.. yup.. all American.

Yet we refer to North, Central, and South America as "The Americas".

Merriam-webster appears to disagree with you. "America" is according to it one of three things:

• either continent (North America or South America) of the western hemisphere
• or the Americas, the lands of the western hemisphere including North, Central, and South America and the West Indies
• United States of America

And that is already the American centric definition of the English word. So yes, "America" in English can mean both, the Americas but also either Southern America or Northern America separately, as a continent. Mind you what definition is listed first. So why shouldn't you say then that Peru is in America than?

https://www.merriam-webster.com/dictionary/america

Worked for a multinational cooperation with co-workers from countries around the globe including south of the USA border, many of these colleagues can and do call themselves American.

You wouldn't say so but they do.