In the 1860 census, my great-grandfather (9 years old) was listed twice--once with his parents and once with another family (I don't know why he was staying with the other people--don't know if they were relatives of some kind). Potentially that could have given Virginia an extra seat in the House of Representatives, but since Virginia was out of the Union from 1861 to 1870, it didn't make any practical difference.
At that time, immigrants to this country were required to sign a loyalty oath and were presumed to be candidates for citizenship. The exceptions were diplomats and mariners. The debate revolved around Indians and slaves. Neither were citizens but the way that they were counted was different. As a compromise, slaves were counted as 3/5th of a citizen. The South wanted them counted the same way as free inhabitants. The North didn't want them counted as all. The compromise gave the South more representatives than did the North's proposal. Before 1860, very few Indians were counted. After that census Indians living with the general population rather than on a reservation. Most reservation Indians were not counted until later. Double counting wasn't rare. Enumerators were assigned a certain households and they counted everyone living in a household. Since the enumerations did not happen on a single day, people in a family might well be counted twice if someone was counted in one household, and then were counted again if they traveled to another household that was counted later. Without computers, they had no way to reconcile all of the returns.
My grandparents, mother and two aunts were counted twice in 1920, at two different addresses