There were thousands of Native American participants in the Civil War. These were primarily Cherokee, Choctaw, Chickasaw, Creek (Muscogee), and Seminole. Small groups of other tribes participated as well including Delaware, Shawnee, Osage, and Caddo. None of the Indian units on either side were "cavalry" but instead "mounted infantry". They brought their own horses and/or acquired them by other means as they were generally not given mounts by the military. The typical Union Indian was issued a regulation uniform and an older vintage military rifle. The Cherokee, Creek, and Seminole in the so-called Union Indian Brigade were trained and drilled in military tactics. Some descriptions of them state they did wear warpaint and added other traditinal "accessories", e.g. feathers, to their uniforms. Their mounts were described as "ponies" (a horse less than 13 hands tall) which made them look distinctly different than the white cavalry units on much larger horses. I have read that the few Osage enrolled in the Union Army seemed to dislike wearing pants and would wear their regulation jacket with either a breachclout alone or with their pants cut into leggings (similar to chaps). However, the typical Union Indian wore his full regulation uniform.
The Confederate Indians were poorly and infrequently supplied, less so than their Arkansas and Texas allies, but did have some uniforms issued to them and a very few firearms. They wore their typical civilian clothing -- typical of whites in the region or an adapted style from combining with elements of traditional dress -- but also butternut, Confederate uniforms if they could get them, and captured Union uniforms, especially later in the war. Capt George W. Grayson of the 1st Creek Mtd Rifles (CSA) said it was common practice to immediately exchange clothing with Union prisoners. They armed themselves as well as they could, bringing personal firearms, often shotguns and obsolete hunting rifles, or what they could pick up on the battlefield or trade for. The famous Randolph Kutz painting of Indian attacking at the Battle of Pea Ridge with spears, feather headdresses and fringed buckskin is pure fantasy.
The descriptions by whites almost always focus on the most novel and/or excentric individuals, not the typical.
It is not accurate to say the so-called Five Civilized Tribes were just like their white neighbors -- some were but not most. In general, they raised some livestock (hogs, cattle), were subsistance farmers (you might say gardeners), and avid hunters and fishermen. Their participation in the war was rooted in hostility between their own tribal factions, particularly in the Cherokee and Creek, but their ultimate purpose was to retain their tribal sovereignty.
Ken