This is really a great question, and one that deserves an answer. Not sure I can provide it, but here is a try:
I cannot remember the exact phrase the historians use, such as Wilbur Cash (The Mind of the South) and Virginia Hamilton (Alabama: A History), but it has to do with a sort of fierceness of spirit that runs through the Southern character and its history. It has to do with such things as Pickett's Charge at Gettysburg, even with the cruelty of slavery, the downright rebelliousness against the North, the obstinance and individualism regarding the Southern way of life, right down to an emphasis upon States' Rights.
The state of Alabama does not have a professional football or basketball team, or a major league baseball team. Yet Birmingham used to claim the title "Football Capital of the South" -- it was right there emblazoned on the upper deck facade at Legion Field. SEC headquarters are still there, even though the SEC championship has moved to Atlanta. If Birmingham had a good domed stadium, they might move the game back there.
It's not just Alabama and Auburn that have college football in this state. The U of North Alabama (formerly Florence State Teachers College) and Troy (formerly Troy State Teachers College) have both fielded tough, championship-level football teams. Bobby Bowden had terrific football teams at Samford U when it was Howard College. I challenge anyone to look around in this country and find a state that has promoted college football as strongly as this state has.
Auburn, to my mind, has trained farmers and pharmacists. If you were to live in or around Foley, Alabama, where WHEP radio is the longest continuous Auburn University affiliate, you might begin to understand something of the Auburn mystique. In the first place, Baldwin County is the largest county in the state, geographically. Roundabout Foley, west, north, and east, are large farms. When asked why WHEP is an Auburn station, the manager's wife said, "It has fed my family." If you go to church in the Foley area, you may see outside a bunch of men after services talking football. If you are a Bama fan, you might as well stay away -- they are talking Auburn football. This is all the more amazing in that Foley is the home of Kenny Stabler.
I think that both farmers and pharmacists would agree that to an extent they build their own little kingdoms. Their lifestyle doesn't lend itself to standing in awe of someone else, and that includes the University of Alabama football team. You don't make it as a farmer or pharmacist standing in awe of someone else. You make it by believing in yourself and your program, by exercising discipline and having faith, a positive attitude in your future and your present. That is basically the same attitude that the University of Alabama has.
When Nick Saban brought "The Process" to Alabama, he brought it to a place that already understands what he is talking about. All he had to do was say the word, and the troops -- that's the fans as well as the players -- understood what he was talking about. It's a certain no-nonsense approach to life, a realization that life can be tough, and that you've got to keep digging, as Bear Bryant once told Jackie Sherrill.
What if anything has separated the University of Alabama as a "football school"? I think it is the understanding that only a championship can bring the plaudits a coach wants to hear, but if he wins that championship, he becomes a legend. They will build museums in your honor and statues of you at Alabama if you win a championship. Why hasn't it happened at Auburn like it did at Tuscaloosa? I find it hard to say. They might have won more national championships under Shug Jordan, but they had to compete with a tradition and a man at the Capstone.