The substance of the role will vary from airline to airline. In airlines I have been in, the Pilot Monitoring does everything except flying; So the walk-around, fuelling and loading supervision, radio, certain selections such as gear and flaps, certain FMGC in-flight programming and changes, flight paperwork, obtaining weather, making position reports etc.
One reason for specifying “Pilot Monitoring”, is probably to reinforce their important role. “Pilot not flying” has negative connotations - suggesting a pilot not doing very much.