Absolutely a "rivalry."
Definitely not their biggest rival.
Besides Vandy, I don't think UT is anyone's biggest rival.
Ask 10 Vols fans who their biggest "rival" is, you might get 4/5 different answers.
Alabama: Auburn
Arkansas: Texas
Auburn: Alabama
Florida: Florida State
Georgia: Auburn/Ga Tech
Kentucky: Louisville
LSU: Miami Dolphins
Mississippi State: Ole Miss
Missouri: Kansas
Ole Miss: Mississippi State
South Carolina: Clemson
Texas A&M: Texas
Vanderbilt: Tennessee
Tennessee: Alabama, UF, UGA, Vandy??????????