With browser's and hardware's support for DRM they could make it impossible if they want to. Basically the OS / recording software sees a blank screen.
I was on live TV recently and wanted to keep a recording for myself, that wasn't just filming the screen with my phone. I first tried screen recording watching the show in my browser in their streaming service. Got a black video. Then I tried their phone app, got a black video. Finally, using my phone but the web page they enabled playback without DRM and I could record and store it. When more devices support DRM they will probably get rid of that fallback as well.