Is it true one democrat and one republican competed in the election of 1860?

Yes, but . . .

There were four parties competing for President in 1860. The Republicans and the Democrats were two, but the Southern Democrats and the Constitutional Union party also were in the mix. Lincoln and Douglas represented the Republican and Democratic Parties. John Bell and John Breckinridge were the candidates of the Constitutional Union and the Southern Democratic Parties.

These were not minor candidates. Breckinridge won eleven states (Basically the deep South) and Bell won three (The Southern border states of Virginia, Kentucky and Tennessee.) Douglas, the Democrat in your question, won only one state (Missouri.)

Abraham Lincoln won eighteen States (essentially all of the states outside the South, California and Oregon -- Maryland and Delaware went to Breckinridge.

One way to look at the election; the Democratic Party split into two and there was a third party as well. Adds up to four parties competing for the Presidency in 1860.