In a 2014 paper, law professor David Schleicher observed that institutional differences between the US and Europe can end up obscuring underlying similarities in the development of party politics.
Some countries have proportional electoral systems that are quite friendly to small new political parties that can grab 5 or 10 percent of the vote. In those kinds of countries, there is an extremely strong incentive for newish political movements to found new parties to represent them.
The United States, by contrast, has political parties that are very “open.”
You don’t need to pay membership fees to vote in a Republican Party primary, and there’s no formal institution like a “shadow Cabinet” that officially speaks for the party while it is in opposition. The party is, instead, a loosely defined network of individuals and institutions that is collectively powerful and permeable. Combine that with first-past-the-post voting, and forming a new political party is a generally unappealing option. What you typically want to do is act entrepreneurially within the structure of existing party politics. This is why Black Lives Matter is pressing Hillary Clinton to disavow old Democratic Party positions rather than running its own candidates for office.
If the US had European-style political institutions, Trump would be leading a European-style “far-right” party. Since it doesn’t, he’s running a GOP primary campaign. But it’s gaining support for the exact same reason that populist parties in Europe are.