Even for the time the 1860 election was an outlier from elections before or after.

Prior to the creation of the Republican Party, Lincoln was a part of the Whig Party which participated in elections in both the North and the South, and was organized around largely economic and political principles that favored the ability of the Federal government to promote economic development, notably through the development of transportation projects such as canals and later railroads. The Democratic Party particularly since the presidency of Andrew Jackson wanted to minimize the size of the Federal government in the economy and leave the bulk of policymaking to the states.

Northern Whigs tended to be anti-slavery and Southern Whigs supported it, as many were slaveholders themselves. The Democratic Party typically represented poorer and more rural whites in both the North (mainly the older states and territories in the Northeast and central Great Lakes states, as the new states formerly a part of the Northwest Territory endorsed Federal policies which supported new land settlement and growth in those states) and in the South, and again there was a split on slavery between those regions.

However, the 1840s and particularly the failures of the Whig Party in the 1850s caused the party to fall apart. Part of it was a split between Southern Whigs which supported the Mexican-American War for the sake of gaining new land for the nation (and new land for slave states) and Northern Whigs like Lincoln who opposed such nakedly imperialist land grabs and also were horrified by the conditions young men faced in the military during the war.

The 1850s saw the real split over the Kansas-Nebraska Act, when Northern Whigs including Lincoln quit the party to form the Republican Party on the platform of containing the expansion of slavery. Many Southern Whigs in the Upper South would vote for the the American aka "Know-Nothing" Party in 1856 which opposed immigration and Catholicism in general. The Democratic Party under the banner of James Buchanan won the election in 1856 on the basis of supporting popular sovereignty in the states over the question of slavery. This idea was popular in the North and South, as many in Northern states were okay with slavery even in the new states as long as their own states would not have to protect slavery.

The 1860 election would not have happened the way it did and arguably the American Civil War would not have occurred without the Dred Scott Decision in 1857. The Supreme Court invalidated the entire concept of popular sovereignty, as they found that the private property rights of slaveowners entitled them to bring their slaves wherever they desired without being deprived of their "property." This decision directly led to the rise of the Republican party in the North, because now Northerners felt as if they did not contain (not eliminate) slavery, it would be extended to their doorstep.

Lincoln was nominated in 1860 on the platform of refusing to extend slavery where it did not already exist (privately Lincoln was in the process of becoming an outright abolitionist, albeit one who still believed recolonization of former slaves to be the best option). This infuriated former Southern Whigs who would run their own candidate under the Constitutional Union Party, largely on the platform of maintaining the Union by any means.

The Democratic Party would fracture at their convention over the issue of whether to support the expansion of slavery or not. Stephen Douglas would be nominated on the Democratic ticket promoting popular sovereignty in the new states, including a possible amendment to the Constitution protecting the rights of states to determine their acceptance of slavery. Southern "Fire-eater" Democrats walked out of the convention when a resolution supporting the extension of slavery to new states whose citizens did not support it was not passed.

The question of why Southerners and especially Southern Democrats opposed popular sovereignty can be explained by the existential crisis the slaveholders of the South faced in the rise of the Republican Party and their platform of containment in the North. Despite Lincoln's claims that he did not support the outright abolition of slavery, the South feared that without the protection of slavery in at least one-third of the states, slavery could be abolished by a Constitutional amendment. Thus why the Southern Democrats ran their own candidate in 1860.

Lincoln won all of the Northern states, John Bell of the Constitutional Union Party won Virginia, Kentucky, and Tennessee, John C. Breckinridge won the rest of the South, and Stephen Douglas won only the state of Missouri.

All of this is largely based on The Fiery Trial by Eric Foner.

Point being that 1860 was a very particular case but is useful for explaining why two parties in the U.S. sometimes splinter, I.e. regionalism.

The divergence of cultural and economic interests between the North and South, which would be seen again in the split between northern and southern Democrats in the 1960s over civil rights.

However when there is a competitive three way presidential election it is typically because of a popular and/or charismatic third candidate taking advantage of splits between or within parties, sometimes having nothing to do with regional interests.

Teddy Roosevelt ran on the Progressive aka "Bull Moose" platform when he felt betrayed by the nomination of Howard Taft by the GOP in 1912. George Wallace ran on the American Independence ticket in opposition to the Democratic Party's promotion of civil rights in 1968. Ross Perot in 1992 pulled away white working and middle class voters who typically voted Republican but were opposed to entitlement reform, immigration, and free trade (sound familiar?), in all likelihood winning the election for Clinton.

Similarly Lincoln would not have won if the Democratic party had been unified in its opposition to the GOP, but without the Dred Scott decision it's likely the Republican party would not have become as prominent without the same factors that led to that fracture.

There are structural reasons why the American political system focuses around two "big tent" parties, mainly first past the post voting and single member districts, but those parties fall apart when there is a wedge between the members of that party coalition.