The relationship varied between tribes and between different factions within tribes. The most relevant areas relate to the Cherokee and other "civilized tribes," which had by that point been removed to Indian Territory (modern Oklahoma).

Notably, the first non-white general in America history was Stand Watie. He was three-quarters Cherokee with one white grandmother. He fought for the Confederacy and was the last Confederate general to lay down arms against the Union.

However, the Cherokees and other natives were not united in support of the South. The Cherokee were divided between a mixed-blood, English-dominant group, and a more traditional group which preferred Cherokee writing and saw themselves as full-blooded. Ironically, the leader of the traditionalist faction, John Ross, was only one-eighth Cherokee by blood, but was considered full blood due to the Cherokee matrilineal tradition. Stand Watie's single white maternal grandmother made him not Cherokee in their eyes.

The two leaders were both slave owners with plantations, but John Ross was generally a supporter of the Union and his four children fought for the Union. Stand Watie, however, was made commander of the Indian Brigade. The Cherokees fought for the South outside of their own territory, the most famous example being the Battle of Pea Ridge. The Eastern Cherokee who remained in North Carolina also organized as a unit and fought as rangers and scouts in the Appalachian Mountains.

Both Southern and Northern newspapers made a propaganda point out of the Natives participation in the war on the side of the South. The Union papers portrayed the Natives fearsome savages liable to scalp white men. They painted the use of such allies to be further proof of Southern barbarism. The South generally portrayed their Native allies as a martial race of Indian braves who were fearsome in battle. Each side generally engaged in stereotyping and exaggerated the novelty of Native in battle. However, other civilized tribes notably did not vote to support the South, and one of the first battles the Cherokee fought for the South was putting down a Creek insurrection.

Here's a link to a site with a picture of a 1903 reunion of Confederate Cherokee veterans:

Cherokee Veterans Reunion Circa 1903

With that historical context in mind, the Confederate-styled dream catcher is not as outrageous as it first appears.