This really just comes down to Auburn and UAB having more alumni who are fans and Bama having more "sidewalk/bandwagon" fans, so of course both AU and UAB have a greater disdain for Bama.
"Alabama fans love Alabama football..." - in my experience this is very true. Very few that I know will follow any other sport except for football. This is true of many schools, I know, as personal sporting preferences differ widely between individuals. But on the flip side, I have way more Auburn fans who follow multiple sports (whether that's basketball, baseball, softball, gymnastics, equestrian, etc.) and who have a fondness for the University as a whole (sidewalk fans or not). Your typical "Bammer/Updyke" couldn't care less if UA was ranked dead last in every academic standard as long as "their" football team is winning.
The only time I get frustrated is when the idiots start trash talking Auburn University itself. You can talk trash about our <insert sport> team but unless you attended Auburn and have an informed reasoning behind your disdain, back. the. f-ck. up. (Normally this applies to those elitist alumni turds from Georgia.) lol