They did not say "Republicans". They said "conservatives".

1800s Republicans were a big business, religious, nationalist party. The Union side in the Civil War was motivated by religion (cc "Battle Hymn of the Republic") and nationalism. Rich men paid to have poor men serve in their stead.

There has been a lot more than "some changes in alignment over the years"—the parties of 160+ years ago are essentially unrecognizable today. Given the nature of how the modern Republican party established its base over the course of the 20th century, this idea of taking pride in a legacy of ending slavery is basically comical, a fiction that falls apart with even a basic knowledge of the history of American political ideology and party alignment.

I don't think it is much more comical than modern Democrats trying to take credit for the civil rights movement. The Democratic alignment has changed from "judge by content of character, not color of skin" to thinking that practice is highly problematic.