This is not true. The airline has the right to ask for it but in practice this is not really done -- or let me rephrase, not consistently done.

I fly intra-Schengen flights at least twice a year. I had to show ID sometimes before COVID but I never had to show ID after that, it actually caught my attention as anybody could have travelled in my place. I do online check-in, drop the bags, go through security, and show the boarding pass. Last time was three months ago, and once again: no ID.

From the top of my mind I can say that I travelled from/to Sweden, Denmark, Germany, Poland, Spain, Portugal in the last few years and I didn't need to present any ID to board the plane.