If one takes a direct flight from one Schengen member state to another, what's the minimum sufficient ID that needs to be provided to board the flight?
It depends on the airline's terms and conditions and potentially between exactly which Schengen countries you are traveling.
The EU regulation requiring id control for flight passengers has been abolished, but there may still be national regulations in effect, requiring airlines to check theirs passengers' identity. Many airlines also require passengers to present a valid id at check-in or when boarding independent of any official requirements and often operate with different lists of documents, which are accepted as "valid ids". In many cases, the id check is probably performed by the airline, just to prevent tickets to be used by other persons than the actual holder.
I have been able to board a flight from Switzerland to the Netherlands with a driver's license from a third country, and Dutch health insurance and bank cards. I didn't plan for it and went up to the counter at the gate as soon as I realized I had lost my ID (this being an intra-Schengen flight, I didn't have to show it before) and after a discussion between all employees present and a phone call to a supervisor, they decided to let me through with a warning that they “wouldn't be responsible if I got in trouble on the other end”. It's just an anecdote but it shows that
- a driver's license is not routinely accepted by every airline as trying to use it created quite a stir (I also don't remember ever witnessing anybody showing anything else than a plastic card looking like a national ID card or a passport-like booklet)
- you might still get lucky sometimes (which I am pretty sure wouldn't happen on a flight outside the EU or even to the UK).
Since there is no passport check on arrival, I went home without problems. I have never been asked for ID at the destination after the many intra-Schengen flights I took in the last few years. But that does not imply that everybody who is already in the area has a right to cross borders or that airlines have to let you board if you don't have a national ID card or travel document.
I do not know if airline/ground handling personnel generally care about your visa status or if they could be somehow held responsible in the unlikely event that the authorities notice they carried someone who did not have the right to enter the destination country (as they can be for other international flights). I never noticed anyone looking at another page than the ID page in passports but obviously that's not very strong evidence.
Importantly, I have perused them many times and I don't think there are any dispositions about that in the Schengen agreement, the convention implementing it, the relevant EU regulations or the Handbook produced by the Commission. To the extent that flying without a national ID or passport is possible, it would be because of local laws, airlines rules or tolerance, it's not something that was explicitly intended as an important part of the Schengen process.
Officially you need travel documents, which are either passport or national ID. These and only these.
Even if you don't need a passport for border checks within the Schengen area, it is still always highly recommended to take a passport or ID card with you, so you can prove your identity if needed (if stopped by police, boarding a plane, etc.). Schengen EU countries have the possibility of adopting national rules obliging you to hold or carry papers and documents when you are present on their territory.
Driving licences, post, bank or tax cards are not accepted as valid travel documents or proof of identity.
Identity checks are done by airline personnel, so you might get lucky some airlines may accept other documents, like residency card or drivers licence, others may not. I've had nasty surprise when RyanAir didn't let me travel with Spanish residency card, which was not a problem for other airlines on same route like Iberia or AirEuropa. But IMHO better safe than sorry.