This a western thing mainly in the States and Canada. When someone is defeated in a competition, mostly combats and fights of some sorts, the loser will be forced to say uncle. Which is a sign of defeat and a way to ask for mercy. The expression started in the 19th century. The expression shows the Republican defeating the Libertarian in a physical overpowering manner. The chauvinist republican would force the Liberal to say uncle, a reference to Uncle Sam which represents all that’s good and American.
Did you know that other countries have national personifications like the USA’s Uncle Sam. Not all countries but most countries that had some kind of revolutions in the country have the norm of national personifications.