Chapotraphouse

They are called that because Andrew Jackson was pro universal white male suffrage (previously there was a wealth requirement).

Before that was the Democrat-Republican party (Andy J was divisive and killed the party) was organized as an opponent to the federalist party who were pro centralization. At the time Members of Federal Congress were appointed by the states so the name Democrat-Republican was somewhat accurate.

They haven't changed their name b/c their branding is still that they are the party "of the people"

Edit: the D-R party was formed during the era of good feelings (after the federalists fell apart), it was the (Jeffersonian) Democrats who were organized in opposition to the Federalists.