In the USA it is possible to fly without an ID?
A lot of people are making general statements, and I'm not sure how valid they are. For example, in my neck of the woods (Canada), I have flown without ID and without passing through security. I would be surprised if the same wasn't true in the US. What I left out: the flights weren't through an international airport and didn't connect to an international airport. Same airport, different flight (one that did connect to an international airport) and passing through security was a requirement. In that case, as well as domestic flights through international airports, ID checks were the domain of the airline.
There are whole catagories of people without "ID" as such, like say underage children or people unable to drive. ID's in the USA have traditionally been either drivers licenses or passports. Many states have added non-drivers license IDs for handicapped, elderly, etc, but AFAIK they aren't particularly popular since those catagories of people don't tend to need them until they suddenly find themselves in a situation needing one.
EU technically doesn’t require government-issued ID to fly either. They often don’t check for ID at all, and in cases where they do, legally any card with your name and photo on it would work for this „identification“. EU generally doesn’t legally require you to carry ID - but they can and will hassle you more and more if you don’t.
I had a friend who flew out of SFO without an ID for many years without much issue. It was much more difficult for them to get back.
Yes.
If you lost your ID while traveling, what would another option be?
Yes, because the federal government can't assume that everyone has an ID, since they don't issue a universal ID. Any attempt to fix the fact that Americans don't have universal federal identification has met stiff resistance from a variety of angles, from privacy proponents to religious nuts who think universal identification is the mark of the beast.
It ties into why we still have to register for the draft (despite not having a draft since the 70s, and being no closer to instituting one than any other western country), and why our best form of universal identification (the Social Security card) is a scrap of cardstock with the words "not to be used for identification" written on it.
So, there's no universal ID, it's illegal to mandate people have ID, and freedom of movement within the United States has been routinely upheld as a core freedom. Thus, no ID required for domestic flights.