This arguement is bullshit

This argument is bullshit

Ok, no.

There seems to be a very popular argument going around lately that over the course of history, the USA has somehow highjacked the term "America" and "American" from its continental neighbours and have ignorantly been applying it to themselves for centuries with no right.

I'm sorry to burst the politically correct bubble here, but yes. They can.
And there's a perfectly simple and valid reason for doing this:

It's in the NAME of the TITLE of the COUNTRY. Literally no other soverign state in the world bears the name "America" on its passport.

Call me fascist, racist, Trumpist, Brexitist or whatever you'd like, but it's why we call citizens of the "Commonwealth of Australia" Australians,
"Federal Republic of Brasil" Brasilians,
"Republic of Colombia" Colombians,"Federal Republic of Germany" Germans,
"Republic of the Philippines" Filipinos,
"Kingdom of Spain" Spanish, and yes... "United States of America" Americans.

The idea that the entire world should now start calling them "United Statesy-ans" (Nevermind that Mexico has already taken Estados Unidos) is like blaming a child for naming their dog "The Dog". All other dog owners angrily get upset when "The Dog" gets 1 million + internet hits in a viral video, even though Fido, Rex, and Spike are all indeed dogs.

Likewise, it's not the average American's fault that their country, birthed out of the British empire some two centuries ago, chose a name nobody else had at the time and rose to such global prominence, rendering the colloquial term "America" to the sovereign we all know today as the "United States of America".

TL;DR - America is in the NAME of the TITLE of the COUNTRY. Of course they have the right to call themselves "America" and "Americans".