Yes the two units are one and the same. The 2nd Cherokee Mounted Vols. were organized in Feb. 1863 from five companies of Maj. J. M. Bryan's 1st Cherokee Partisan Rangers and a like number of extra companies of Stand Watie's 1st Cherokee Mounted Vols. W. P. Adair became colonel of this Regt.
I can find no evidence in the service records of the 2nd Cherokee Mtd. Vols. of an Isom Edwards. But, that doesn't necessarily mean he didn't serve in that unit. The service records for the Indian Regt's after July 1863 have not servived. So, he could have served after that date and there just be no record of same. There were literly hundreds of men (and boys) from the counties of Benton, Washington and the other border counties with the Indian Territory that served in the Indian regiments, especially the 2nd Cherokee Mtd. Vols.