It wasn’t until after TWA flight 800 that the U.S. government began requiring passengers to show ID. President Clinton wanted to be able to announce something to look like he was doing something after the event, since initially there was speculation that the explosion was terrorist-related rather than an accident.
Airlines have wanted customers to show ID for a long time because they don’t want their tickets to be transferable. Part of making sure every passenger pays as much money for a seat as possible means ensuring that cheap seats can’t be resold. Otherwise the airline wouldn’t be able to sell last minute tickets for the last seat available for a premium – other passengers would just resell their tickets at a profit instead.
You can still fly without ID, however. The TSA has a procedure to let you clear security checkpoints without one. People lose their drivers licenses all the time.
If you don’t have your ID, it’s best to check in online, with your airline’s mobile app, or even at a kiosk. Agents at the ticket counter will ask to see ID.
American Airlines has a procedure for passengers to fly without ID, just like TSA does. However they’re cracking down on the ID requirement when you get a boarding pass from an agent at a gate, trying to ensure that gate agents insist on seeing IDs the way that ticket counter agents do.
I don’t actually recall a time when an American Airlines gate agent didn’t ask me for ID when issuing me a new boarding pass. However this must have been common enough for the airline to announce it as a policy.