Yes, I have experienced this quite a few times on intra-Schengen flights.
As said in the link from the comments Lufthansa probably is the most prominent case due to the extreme amount of automation, but I have flown with a number of airlines and from/to a number of airports and can tell you it depends on the airport/airline combination.
Routes I have flown within the last two years without having passports checked (note that routes listed are strictly one-way, I will list return flights separately if they were without checks):
PSA-CPH Norwegian, BCN-PSA Vueling, PSA-BUD Ryanair, STR-FLR Meridiana/Air Berlin
Note that I have been checked on other routes/return legs on Vueling, Ryanair, Meridiana - and I can't judge in the case of Norwegian.
Thus the answer to your question is - yes, some airlines, on some routes do, but passport checks are still very much the norm, even within the Schengen area.
For the US specifically, the Transportation Safety Authority website states:
Adult passengers 18 and over must show valid identification at the
airport checkpoint in order to travel.
And has a list of acceptable forms of ID which includes obvious things like passports and residence permits, but also drivers license and some traveler cards - full list here.