Usage and meaning of "plus"

Solution 1:

Plus is Latin. Means "more" but in the additive way.

Non plus ultra = No más allá. No further than that.

1 plus 1.

When used in names, means "best", exaggeration, some kind of "push up". I'm just recalling this so you surely should check the Latin original term and the usage in English today.

In your examples. American Plus = American Best, American Top.

As with many Latin words, the translation is subjective to the context, but when used in branding, is just a "cliché" to say you're top.

Solution 2:

I'm having trouble assigning any meaning to America Plus or American Plus. I kind of want to add another word: America plus Europe equals [something], but that doesn't really make any more sense, either.

Besides the mathematical sense ("two plus two equals four"), you see the word plus used in the combination plus-sized "needs to shop for clothes in the big & tall section"; as a synonym for advantage: "the pluses outweigh the minuses"; and as a coordinating conjunction meaning something like also: "He had an overbite, plus he stuttered, so he didn't have many friends at school."