During the American Civil War, which began in 1861 during Lincoln's presidential term, the image of an elephant was featured as a Republican symbol in at least one political cartoon and a newspaper illustration.
Soldiers used the term "seeing the elephant" as an expression meaning experiencing combat, and Nast later translated the animal into his political cartoons portraying the Republican party.
In 1874, Nast used the elephant in Harper's Weekly magazine, and that is when the elephant started to take hold as the Republican party's symbol and is now synonymous with the party.
Two different animals. Two different paths. Two different parties: the Democratic donkey and the Republican elephant.