Why are the 2 main parties in the United States called the Democrats and the Republicans? The US is both a democracy and a republic, and neither party's platforms seem to be built around the supporting or opposing one system over the other. So where did the names come from?
The names 'Republican' and 'Democrat' have symbolic meaning from the eras in which they were chosen, and are retained to the modern era — where they are no longer entirely meaningful or relevant — mainly because of path dependency. It is extremely problematic to change the identifying name of a group mid-stream, because many people track groups specifically by name. Some groups do it intentionally in order to escape negative associations (e.g., the way Blackwater Security Consulting changed its name after the Nisour Square Massacre so that it could continue operations without protests), but for an explicitly political organization changing names could cause massive losses of recognition and support.
As to the origins of these names, the first US party system was composed of the Federalist party and the Democratic-Republican party. The first name was meant to suggest a strong central government and closer ties to the British Monarchy (similar, perhaps, to modern-day Canada or Australia), while the Democratic-Republican name was meant to imply a platform that focused on the American people (demos) and the independence of the Republic. The Federalist party did not last long, and when it collapsed the Democratic-Republican party splintered into the Democratic party (who held that the President and Executive branch should be the primary branch of government) and the Whig party (who held that power should inhere in Congress). I'm not certain where the term 'Whig' came from; the natural assumption is that it was a reference to some characteristic of the British Whig party, but I don't know the reference off-hand. In any case, the Whigs succumbed to internal divisions in the run-up to the Civil War, primarily over the issue of slavery and abolition. The Republican party formed out of the abolitionist elements of the fractured Whig party, with the term 'Republican' revived to indicate its focus on the integrity of the Republic and the ascendency of Federal power over the States' Rights model that the Democrats (at that time) supported.
That was the Third Party System, and in the intervening century and a half the Republican and Democratic parties have undergone several polarity shifts, dramatically altering and even reversing their foci, platforms, and policies. Party goals no longer align with the symbolic intent of the original names; the names are retained primarily for political continuity. But the names do reflect early issues of how power should be distributed between and within the federal and state governments.