I have the same problem here On a Mac mini M1.
It was working fine with Big Sur and started to fail after being upgraded to Monterey.
After several hours of tests (first thinking that I forgot something myself), I'm fed up and I give up.
Note that I Also have a Mac Air M1. When I try to copy screens to it from the Mini, The second Screen of the mini start to work, displaying a copy of one the other screens (1st screen of Mini or screen of Air, depending of the display parameter I try).
So there is something looking like a dysfunction with the share screen capability.
The only solution from now seems to be to downgrade to Big Sur. It's such a shame ! Just when I started to thing that MacOs was not so bad.
When I read that the only proposition is "Disconnect the cable from your external display from your Mac, then reconnect it. If the cable from your external display doesn't connect to the ports on your Mac, you can use a Thunderbolt 3 or USB-C adapter.", I'm half afraid, half furious, and finally very sad.