Well the RATs have Jefferson/Jackson dinners every year as a fund raiser...much like the GOPs have a Lincoln Day dinner.
They’ll say things like that, because, hey, Jeffersonians became Jacksonians, and both were fir agrarians and against Yankee mercantilists, and so on. But it’s false history. Jacksonian democracy was not Jeffersonian. Read “The Era of Good Feelings” by George Dangerfield, for reference. Jackson is the real foundation of the Democrat Party.
Republicans might as well claim Whigism as its true foundation, and that would be more ideologically accurate. But they don’t, and just as well. Because Republicans weren’t Whigs even if Lincoln was a Whig before he was a Republican, for instance. There are reasons the former party failed and the latter dominated Washington from Lincoln to Wilson.