sad to hear and thanks for the infoNeither gets completely powered off unless you unplug them.
But, shutdown puts both the pc and the monitor into a very low power state that is very close to powered off.
Sleep will also do that and sleep to ram is much faster to sleep and wake than power off/on.
That was "possible" back in '80s and early '90s, when computer' power buttons were really toggle switches which controlled the mains power to computer' PSU, and monitors were commonly powered by slave cable plugged into computer' PSU.
These days, you'll need additional hardware like this USB-controlled power strip