I'm not the author but If I have to guess, author must have recorded the phone screen ( using something similar to https://play.google.com/store/apps/details?id=com.hecorat.sc... ) and desktop screen, overlapped the phone screen video onto the desktop using video overlay feature during post production.

You can use adb to screenrecord. (I have never tried it though)

https://developer.android.com/studio/command-line/shell.html

EDIT: I think this actually does not answer your question. Sorry.

Are you sure that's his phone and not Android Studio's Android Emulator (https://developer.android.com/studio/run/emulator.html)?

I didn't. Recorded videos on both the phone and the computer(a VM actually), then put it together with.. I don't remember what. Some video editor available in the Ubuntu repos.