I'm sure that others could tell it more accurately and with more detail than I can do, but in general the boots are a symbol and riding the horse is an honor.
The military side of Texas A&M naturally uses seniority for status among the corps of cadets. At the lowest level are Freshmen/Fsih (or new transfers into A&M).
In the military arrangements at A&M, there are two elite groups. The first is the Cavalry and the second is/are the Ross Volunteers.
To discern "rank" or seniority, there are subtle changes to the Texas A&M military uniform as a cadet/student advances from Freshman to Senior. The Freshmen have the shortest haircuts, for instance, while Juniors get to wear white belts (I think) and Seniors get to wear boots.
But they aren't just any boots. They are nearly knee-high, dress, mirrored-finish boots.
In the cavalry, cadets start by scooping up the excesses of the horses (a hilarious sight to see at football games as the Freshmen follow the horses with brooms and shovels to clear the deposits - with uniform military precision). Riding one of the horses is pretty well forbidden until much later in the cadet's career at A&M.
As Juniors, the cadets anxiously await the special day wherein the Seniors yield and permit them to finally wear their polished boots. Many a fine Southern lady has disdained her Texas suitors until they have worn those boots for the first time, as being escorted to a Texas A&M football game by a man who wears the boots is far more desireable than merely attending the event with someone who has no such display of power and status, it seems.
Moreover, those in the cavalry anxiously await the day wherein they can finally ride their horse, preferably to/at an A&M football game.
The Aggie who has both the boots and the horse is close to the pinnacle of A&M campus status. While the horse won't make every class, however, the boots will (and therefore the boots are also a found target for the brave practical jokes by the lower classmen to be played upon the Seniors).
Not everyone gets to wear the boots. Fewer still earn the right to ride the horse, and on campus, those are enormous symbols of respect.
To do them both, your friend put in the time and maintained his honor for years.
Whoop!