I don't know every countries rules either, but I live in France and take dozens of domestic flights in France and dozens more within Schengen each year. If any of those flights didn't require an ID I don't remember it. I do clearly remember people around me showing their national ID and I always have to show mine. I fly in and out of Spain often and I know they require ID's for Schengen flights. I haven't been able to get the KLM check-in machines to work for me so I'm not sure about NL.

Also, I know you have to show ID's for LH flights to FRA from France.