Earlier quoted context omitted.
LOL, but why aren't french-speakers from Quebec called "latinos" also? It makes no sense.
Language and nomenclature does not develop rationally. It's fair to point out that you find it jarring, but it's so embedded in the western hemisphere that your appeals to sense aren't going to change anything.
I am in the "western hemisphere" (by any possible definition), and this nomenclature is not by any means embedded in my culture. In high school, we learned latin as the language of our ancestors, we studied latin civilization and latin culture as part of the ancient Roman empire, etc. I remember being taught about our latin culture of southern europe of romance-speaking peoples, and they compared with the germanic culture of the north. At home, my father pushed into me the (of course, ridiculous) stereotypes of the serious, hard-working germanic peoples of the north, which are probably more intelligent but they have less creativity than us latins. I can call myself a "proud latin".
It does not seem that this usage of "latin" that you mention is particular to the western hemisphere, but rather to the U.S.. Moreover, the term "latin america" was precisely introduced to refer to the french-speaking colonizers of the continent.