You guys are naming programs that exist in the same state, which is way different. Also, Auburn has multiple national championships in football. They are not as good as Alabama, but they are still an elite program, which Providence is not.
Obviously "rivalry" is subjective, so Providence being a rival completely depends on how you define that term. For me, a rivalry exists based upon playing many very important games over some extended period of time where the results of those games have large implications on the historical success and perception of each program involved. So Providence is not a rival to me, because we don't have that history. I consider St. Johns, Pitt, Louisville, and obviously Syracuse much bigger rivals.
If someone's definition of a rival takes into account fan interaction on the interwebs, or physical proximity, etc. then Providence would be a rival, and that's fair. It's just not how I would define it.