Evidently some cavalry units made the effort early in the war:
The horses were sorted according to color, the intention being that each unit should have but one color, as near as practicable. Thus, as I remember it, troop "A" had bays; "B" browns; "C" greys; "D" blacks; and so on. This arrangement did not last long. A few months' service sufficed to do away with it and horses thereafter were issued indiscriminately. The effect, however, so long as the distinction could be kept up, was fine. It was a grand sight when the twelve hundred horses were in line, formed for parade or drill in single rank, each troop distinguishable from the others by the color of the horses. (James H. Kidd, Personal Recollections of a Cavalryman with Custer's Michigan Cavalry Brigade in the Civil War (online), 6th Michigan Cavalry)
But here are two later examples, from July 1863, during the Gettysburg campaign:
John Collins saw his own regiment – I could distinguish the companies by the colors of their horses, and knew the order of the squadrons in the line. The black horses of troop C and light bays of H formed the first squadron, the sorrel horses of E and the dark bays of G formed the next, and so on. The troops changed squadrons often to suit the seniority of the captains, and the squadrons changed positions in the regiment for the same reason, but the combination of companies before me now had been that of the regiment for a week at least. (John Collins, 8th Pennsylvania Cavalry, Battles and Leaders of the Civil War, vol. 3)
A soldier described the 1st West Virginia Cavalry at Emmitsburg on July 1 as consisting of "about 600 men, all mounted on the finest black horses, many if not most of the men from Ohio." (William B. Southerton, Papers, 75th Ohio)