I've been having these same issues with trying to share my iOS devices to the Mac during Zoom sessions. The problem is bluetooth and trying to connect iOS devices to your Mac. The source of the problem is NOT the M1 Mac but the OS, Monterey. I'm having these problems on both an M2 MacBook Pro AND on an Intel MacBook Pro, both running Monterey. I KNOW it's a problem with Monterey 12.1. I'm not sure about the earlier version of Monterey. Anyone who is running Monterey 12.1, try connecting your iOS devices to your Mac via bluetooth. Your devices either won't connect or will connect and then immediately disconnect.
Previously, running Big Sur, I had no problems whatsoever with the old Intel MacBook Pro and Zoom. I was able to share all of my iOS devices. Since updating the Intel Mac to Monterey, specifically Monterey 12.1, I can no longer share iOS devices while running Zoom on the Mac. The other Mac, my M1 MacBook Pro has been having the problems pretty much from the beginning I believe.
I've talked to Apple Support several times and been to the Genius Bar at the Apple Store. Both places have confirmed the problem of connecting iOS devices to the Mac when running Monterey 12.1. They suggest I use Airplay. But as everyone probably already knows, this new version of Airplay forces your iOS device to take over the entire Mac screen (basically your Mac simply mirrors the iOS device's display). Unfortunately, this doesn't work well with Zoom, and if somehow you do get your iOS device to show up on the screen during your Zoom session, you can no longer control your Zoom session while the iOS device is showing. And if you record the session, you'll end up with a recording of a black screen not of your iOS device.
In my opinion, this is a serious problem and Apple needs to fix the Mac OS, so that we can still use bluetooth when needed to connect our iOS devices. I guessing that a lot of online streamers and people teaching online courses are going to be pretty upset when they try to use Zoom on their Macs and find they can't share the screens of their iOS devices during a session. And there are other situations where a bluetooth connection to an iOS device is needed -- e.g. using your MacBook Keyboard to type on your Mac and an iOS device (e.g. 1Keyboard is one such App).