Very rarely are there absolutes in history. "Sole" indicates the exclusion of all other purposes. And the military does shuffle units around as necessary. Custer's unit in fact spent two years doing military occupation duty after the Civil War (1871 - 1873). And if war with Mexico was necessary during the time period it is likely 7th Calvary would have been dispatched.
But we can safely say Indian warfare was a primary purpose of organizing the 7th Calvary. It was organized in 1866, in the middle of Kansas, more than 900 miles from any international border. It then spent 28 of it's first thirty years in existence in Indian warfare.
Not true at all.
The post Civil War U.S. Army was dramatically reduced in size from its massive wartime scale, shrinking to just a few tens of thousands from its peak of over 1 million. Its responsibilities were myriad. Arguably paramount was the protection and stability of the 36 states that comprised the United States post Civil War.
This included not just occupation/reconstruction, but rebuilding of the Army itself. Few enlisted men chose to remain in the Army once the War was over, necessitating replacement even in light of the sweeping reduction in size of the force. Essentially a virtually new force needed outfitting and training, one which would be far more cavalry-oriented than ever before.
Additional duties ran the gamut from securing the Mexican border to protecting American interests in the territories that were not yet states. This included such activities as guarding traffic on the Missouri River, escorting railroad surveying parties for the various cross-country building projects that were underway at the time and building and maintaining a network of military outposts both along the established frontier and in locations that served to support their mission within the territories. The latter duties were focused on securing the emerging common cross-country travel routes and lines of communication while also reconnoitering the parts of said territories that were essentially unknown to whites. This mission of course included contact with Indians, ranging from conversational to confrontational.
In the late 1860s and 1870s at the peak of the Indian Wars there were only around ~25,000 soldiers on the entire U.S. frontier. Roughly 15,000 of these were in the Military Division of the Missouri, a sphere of responsibility commanded by Phil Sheridan that ranged north to south from the Canadian border to the Texas line and east to west from the Mississippi River to beyond the Rocky Mountains, an area comprising over a million square miles.
Formed in 1866, at various times during their existence the 7th Cavalry was charged with typically wide-ranging responsibilities. Rarely were all companies serving in the same location as a complete entity, one of the only times this occurred in the field being the Battle of the Little Bighorn. They established military posts in Kansas, what is now Oklahoma (then called Indian Territory), the Dakotas and Montana. They did reconstruction duty throughout the deep South, acting in a police capacity. They were charged with procuring mounts for the Army in the horse country of Kentucky. They escorted surveyors for multiple railroad lines. They manned a remote outpost near the Canadian border in the Dakotas. They explored the Black Hills in 1874, an expedition that served to confirm earlier reports of gold having been found therein.
Did they also fight Indians? Indeed they did, but far less often than common misperception holds. Their ten-year existence up to and including the Sioux campaign of 1876 included the Battle of the Washita, the Battle of the Little Bighorn where Custer and more than 5 companies of the 7th met their demise, and four other encounters which were akin to skirmishes with few casualties suffered on either side.
Less than half of the officers of the 7th Cavalry at the Battle of the Little Bighorn had fought Indians prior to that engagement. At least 30% of the enlisted men were raw recruits who had not even been fully trained or in the field let alone in combat prior to the 1876 campaign.
Sources: Custer, My Life on the Plains; Utley, Cavalier in Buckskin: George Armstrong Custer and the Western Military Frontier; Wert, Custer: The Controversial Life of George Armstrong Custer; Marquis, Keep the Last Bullet for Yourself; Hutton, ed., The Custer Reader; Donovan A Terrible Glory; Welty, Kansas Historical Quarterly, The Policing of the Frontier by the Army, 1860-1870; Robinson, A Good Year to Die: The Story of the Great Sioux War.