None of us have insider information and therefore we are ALL guessing. But it's more educated guessing. Nobody knows everything.
It's a Catch-22 scenario due to a bug in 15.0 that allowed the user to begin the process of setting up iPhone Mirroring but then it failed because the iMac 2019 lacks a T2 Security Chip and is incompatible. It was then left in this incomplete state. When 15.0.1 installed it added the alert about not being compatible. But now the user cannot reach the iPhone Mirroring Settings in order to click the Reset iPhone button which will factory reset iPhone Mirroring. Once reset iPhone Mirroring resets back to the state
The only other thing that can cause the require password immediately and lock it down you can't change it; would be due to FileVault and MDM (Mobile Device Management). User should check there are no Configuration Profiles installed in System Settings -> General -> Device Management. There should be nothing in the list of profiles on a retail non-managed Mac.
Another thing, occurs to me. When you open iPhone Mirroring and receive the error that it is not compatible. DO NOT CLICK on the message. Instead click on the Menu bar where you hopefully see iPhone Mirroring. You might need to CMD-TAB away from the alert dialog to set focus on iPhone Mirroring so it appears on the menubar.
It really depends on if iPhone Mirroring is open and stays open when the alert dialog appears. Clicking on the alert may quit the App if the alert comes from the App. Now if the alert comes from elsewhere like macOS itself then it might exit iPhone Mirroring when the alert pops up. If you are lucky that won't be the case.
I did find another solution that worked for others:
Quit iPhone Mirroring then
Open Terminal:
# Backup the Preference file to your desktop
defaults export com.apple.ScreenContinuity ~/Desktop/ScreenContinuityDefaults.plist
# Delete an entry from the file
defaults delete com.apple.ScreenContinuity onenessPairedDeviceID
# Recover / Revert in case something doesn't go right
defaults import com.apple.ScreenContinuity ~/Desktop/ScreenContinuityDefaults.plist
Executing this seems to do the same thing as Reset iPhone button in iPhone Mirroring Settings. It entirely reset my iPhone Mirroring. Fingers crossed, this works for you.
The time consuming absolute last resort would be to backup your data, burn a bootable installer to a flash disk. Hold ⌥ Option while powering on (Apple Silicon hold power button). Boot from the Install macOS Sequoia. Erase your internal SSD, name it Macintosh HD and then re-install Sequoia 15.0.1. Then if Time Machine let it restore everything. If the problem carries over after you restore the backup, then you'll need to rinse and repeat but exclude System & Network and perhaps Other Files & Folders which may contain the settings for iPhone Mirroring.
