Even as the two-party system has remained rather stable over the course of American history, new political identities have always been proliferating in response to social, cultural, and economic trends and events.
There were the rowdy “workies” and “Loco Focos” of Jacksonian America, who arose in response to the dislocations wrought by the “Market Revolution” of the early 19th century and were more or less incorporated into the Democratic Party’s fold. Then, of course, there were the antebellum, anti-slavery advocates who spanned both the Democratic and Whig Parties (remember that antebellum Democrats weren’t always purely pro-slavery; the Free Soilers were largely Jacksonian Democrats), and then opted for third-partyism, first unsuccessfully (the Liberty Party) and then successfully (the Republican Party). Later on, there were the “Progressives” of the early 20th century within both the Republican and Democratic parties, and even formed their own third party in support of Teddy Roosevelt’s 1912 presidential bid as the nation sought to grapple with the social and economic changes brought on by industrialization.
There are more recent examples, too. The movement conservatives that grew out of opposition to the New Deal offered a unique brand within the Republican Party before overtaking it during the Reagan era. There were also the Arthur Schlesinger-style mid-20th century liberals (like the Americans for Democratic Action) who sought to uphold a left-of-center politics within the Democratic Party while distancing themselves from Progressives and Communists.
Some of these political identities spanned the two major parties, some formed third parties themselves, and some overtook one of the two major parties. While they differed in terms of their long-term electoral success, they all arose in response to potent forces in American life to which the two-party establishment, to some degree, was failing to adequately respond.