@n80, if your comment was made in regard to the names I mentioned (it may not have been), I named the band, then gave details about it’s members. Bands have members, right? ;-)
If The Band is eligible, what makes them SO unique is that they are a self-contained "unit" (writing their songs, singing them, and playing the instruments) that is at least as good as any band hired by solo artists for accompaniment. If you are "just" a backing band that provides accompaniment for a solo artist, you have to be a REALLY good band. Rock 'n' Roll bands are known for their songs, sound, and/or style, not necessarily for their excellence as a performing band, in the old sense of the term. For instance, The Heartbreakers made excellent music, but provided poor accompaniment for Bob Dylan. Not as bad as The Dead, of course ;-). On the other hand, The Band were an excellent band in the sense I am speaking of, as good as any group of studio musicians, many of them the best in the world.