When ever I checked in at the special assistance counter (when travelling with partner), we had to show ID.
And I also have had the experience several times not being able to check in at Kiosk - sometimes but not always asked for ID at premium counter.
As an aside recently in Hawaii had two flights to take where the airline issued a ticket with incorrect spelling of my name - was worried TSA was going to pick it up and not let me through security. But they are much more relaxed at HNL and ITO, than at LAX.