A Jewish bagel weighs about three times as much as a non-Jewish bagel, and is about three times as dense. It doesn't have the airy, bread-like consistency that a non-Jewish bagel has. As my Jewish friends like to say (paraphrasing), "If I can throw it at your head and risk knocking you out, then it's a Jewish bagel."
I've never had a bagel that was "airy", but I've also never had one that was made without yeast. They're normally pretty chewy, but definitely not unleavened. You boil and then bake them.
No, they're referring to real bagels (chewy, with a shiny exterior) as "Jewish", as opposed to fluffy-white-bread-roll-with-a-hole that is sometimes marketed as a bagel.
I am certain that "chewy with a shiny exterior" bagels that I get at a bakery are made with yeast. They aren't some sort of secret or rare thing any place I've been, from New York to Arizona.
The mass-produced-in-a-factory-with-lots-of-preservatives bagels (e.g. Lenders) are very different, but also obviously made with yeast, and they don't really resemble fluffy white bread IMO.
When I go to my local grocery store in the northeastern US (the chain is in fact owned by a Jewish family), both kinds are available, the one in the bakery area and the other on the shelves with rolls.
Talking about these as though the dividing line was Jewishness or the use of yeast strikes me as both weird and incorrect.