Does any airline in the world allow air travel without government identification?

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.

In the UK, I have travelled a few times with Loganair - Scotland's regional airline. Not once was I asked for an ID, not even when I purchased the tickets in person at the airport in Kirkwall. Note, of course, that their island hopper planes sit 9 passengers and don't even have a dedicated cockpit, so the pilot simply sits in the front row of the plane. Maybe security isn't really their biggest problem.

It's one of these aircraft:

If flying with British Airways (BA) domestically within the UK, you can use your British Airways Executive Club card as ID. (BAEC = the BA frequent flier program). If you're not a member of the BA Executive Club, you have to show some form of photo ID.

I believe that BA still suggest you carry some photo ID with you when doing this, eg in case of being asked by police, but you can (and I have) flown with them domestically just using your loyalty card as ID.

Virgin Australia recommend that you take government ID with your when flying within Australia, but were happy to accept my Virgin Atlantic frequent flier card as proof of identity. Like the BAEC card, neither has any photos or anything like that on it.

I have taken six domestic flights during my latest trip to Japan with three different airlines and wasn't asked for my passport (or any other ID) for any of them. It may be related to the fact that Japan does not issue national ID cards as some other countries do (the closest thing is a driving license, but it is expensive and troublesome to get). The airlines don't state this as a requirements on their websites either (see ANA, JAL).

(However, all foreigners in Japan are required by law to carry ID at all times—passports for tourists and residence cards for long-term residents.)