The Republican party was renamed the Union party for the 1864 election. Therefore, Lincoln also served under the Union party label. For Washington’s initial election, political parties were not in existence. He became associated with the Federalist party after he was in office.

The Whig Party was a political party active in the middle of the 19th century in the United States. Alongside the Democratic Party, it was one of the two major parties in the United States during the late 1830s, the 1840s, and the early 1850s, part of the period some scholars describe as the Second Party System. (https://en.wikipedia.org/wiki/Whig_Party_(United_States))