There's a bit of a debate about this. On one hand, Colt did supply his Patterson revolvers with a spare cylinder. The Prussians also made up some holsters for Colt revolvers that had a pouch for a spare cylinder, and similar US pouches exist. A soldier could certainly have a gunsmith fit a second cylinder to his revolver, regardless of how it came when he bought it. So, you can't say it never happened.

However, that soldier could not grab the spare cylinder and reload the gun in the same manner as someone now ejecting a magazine on, say, a 1911A1 Colt automatic, and quickly loading with another. The revolver had to be taken apart, with a wedge knocked out and the barrel and loading lever assembly pulled off the arbor, allowing the cylinder to be removed. Then the gun had to be reassembled with the new cylinder, and the wedge knocked back in. This would not be very practical for anyone in a hurry ( especially anyone in a hurry riding on a horse). So, yes, it would be possible. But, it's pretty safe to say it would not be something that could be expected to take the place of having two pistols: and there are real photos of Civil War solders sporting two pistols.