I fell in the same situation a year ago. I showed my EU driver's license at check in desk but was refused checking in because the clerk stated that driver's license does not show nationality. Then I gave her my national ID card and everything was ok.
First, the question "please show me your passport" is probably a mantra as stated by @Lassi. Just like when you are checked tickets in a bus but officers ask you to show pass (in the sense monthly pass used by frequent commuters).
Anyone can get a driver's license in any EU country regardless of his/her nationality. So the driver's license does is surely not accepted because does not prove EU citizenship.
Now to Schengen agreement: while it is commonly called to be "no borders" (and no checks) between countries, mind that freedom of circulation in Schengen space is granted only to Schengen citizens. So you basically have to prove the airline that you are a Schengen citizen to travel the Schengen area.
That is what was told me.