I refer you to Vernon Crow's "Storm In the Mountains,Thomas' Confederate Legion of Cherokee Indians and Mountaineers"

The picture you depict is titled in the book "Cherokee Indian veterans of Thomas's Legion" at the 1903 New Orleans Confederate Reunion.

Further, is a photo of Col. Thomas's grand-daughters holding the battleflag of the Infantry Regiment which had been donated by the family to the
Museum of the Cherokee Indian.

There doesn't appear to be anything distinct about these standard Confederate battleflags which would indicate association with the Legion.