As a matter of historical continuity of the organization, yes, today’s GOP is the party that elected Lincoln in 1860. (In 1864, Lincoln was technically the candidate of the Union Party.) Jefferson’s Republican Party, on the other hand, became today’s Democrats.
In the aftermath of the war, the Republicans had the power to denounce all Democrats and Southerners as “traitors”, and, consequently having a monopoly on power, proceeded to go straight to Hell. There were only about two honest men in the Grant administration, and the 1876 presidential election was stolen outright by wholly unconstitutional acts of the Republican congress.
Corrupt governments, of course, are always the friends of the rich, so the Republicans became the party of capitalism, which was already far stronger in the North.
Eventually some reaction set in, and the Democrats began to pick up the support of farmers and labor, to the point where they could win the White House if the Republicans made a big enough mistake. That’s still largely the situation.
However, there was no clear “conservative” / “liberal” division yet. The Republicans were definitely the party of the rich, and the Democrats were definitely the party of poor whites, but the South continued to stick with the Democrats, partly because they could not stomach the thought of joining the party of Lincoln, and partly because the South was poor. (I have a survey of US education in the 50’s that remarks offhand that the book would be discussing only the North, because, apart from a handful of good universities, the South had to be regarded as part of the Third World.) Southern blacks were another matter, but almost none of them could vote. There was a clear tension in the situation; the 1948 breakaway “Dixiecrats” were one symptom, and it is almost unimaginable to people who don’t remember the time that George Wallace, for example, was, on all points but one, a strong liberal.
One other oddity was that, due to a long tradition of Democratic corruption, the Republicans were the de-facto liberals in New York City. As a general rule, they could control the city if and only if they ran a fusion campaign with NYC’s Liberal party (created by anti-Marxist breakaway Socialists).
In 1964, everything changed. Barry Goldwater’s “Southern strategy” suddenly made the Republicans the party of segregation. We are still living in the political world he created.