If you want to go far enough back into history. How about the Browns? They can say that too. So can the Skins.
The Cowboys will not suffer that because of a term coined early on to try and explain the attraction for a team that had really not accomplished much and I don't know who gets credit for it but it is still in play today, "Cowboys Mystique". That's what created America's team. I do not know how to describe it but can attest to it that is was there back in the 60's. I was a Cards fan and they became my co-team immediately and exclusive in the Ice Bowl.
They used to also have a label that the players really detested when compared to the Browns or Packers, the "finesse team". They pretty much shut that down for the Packers fans in the Ice Bowl. Guy I used to work with was from GB and he said the talk of the town wasn't the Packers the next day, it was the toughness of the Cowboys. I remember what the great GB announcer, Ray Scott, said about them after that game. The Cowboys could play in the black and blue division.
You are right, but not just about the Bengals. I don't care how many rings NE wins or PIT or the titles in GB, there is no NFL franchise like the Dallas Cowboys throughout it's 100 year history. Earned or not, doesn't matter, it is still there.