Lincoln WAS a Republican, but in 1864 the "National Union" Party was formed. It was comprised of Republicans AND Democrats who supported Lincoln's policies--continue the Civil War.
The Democrat who ran for President was George McClellan--who wanted to END the War and let the South go "in peace".
To create a balanced ticket, the National Union party nominated Lincoln--a Republican and Johnson--a Democrat.
Therefore, when Lincoln was Assassinated, he was succeeded by a Democrat--who faced a VERY hostile Republican Cabinet and Congress.
"I say YOU'RE the CUTEST one. No, I say YOU'RE the CUTEST One. And we go on like that from dawn to three."