Which is the only American coin to be "two-headed", with a portrait of a famous person on each side?

Most double-headed coins are mint errors

However, the first general circulation regular-issue United States coin that features the same person on the obverse and reverse is the Lincoln Memorial Cent, which was produced from 1959 to 2008. The obverse shows a right-facing portrait of Abraham Lincoln, the 16th US president. Until 1959 it was paired with a wheat ears reverse motif. Sculptor-engraver Frank Gasparro designed the Lincoln Memorial reverse, the Washington, D.C. landmark that opened in 1922. The design includes the seated 11-foot statue of Abraham Lincoln which is a feature of the memorial. Therefore this coin features a figure of Lincoln on both the obverse and reverse.

https://www.pcgs.com/news/lincoln-memorial-cents-fill-eyes-with-double-vision

If the above is the criteria then there are several US Coins that depict portraits of people on both sides
the 1999 New Jersey Quarter has George Washington and, on the reverse, shows him standing in a boat crossing the Delaware river. The 2003 Illinois State Quarter again shows George Washington and, on the reverse, shows a young Abraham Lincoln. The 2003 Alabama State quarter show George Washington and, on the reverse Hellen Keller, just to name a few. There is also the American Women coin series

https://www.usmint.gov/coins/coin-medal-programs/50-state-quarters/new-jersey

https://www.usmint.gov/coins/coin-medal-programs/50-state-quarters/illinois

https://www.usmint.gov/coins/coin-medal-programs/50-state-quarters/alabama

https://www.usmint.gov/coins/coin-medal-programs/american-women-quarters/maya-angelou

https://www.usmint.gov/coins/coin-medal-programs/american-women-quarters/sally-ride