Is iOS 16 compatible with MacOS Monterey?
Yes, an iPhone with iOS 16 compatible with MacOS Monterey.
but now a dialog pops up saying "A software update is required to connect to <iphone name>"
Are you referring to this? -> If you see "Software update is required to connect to your iOS device" on your iPhone, iPad or iPod touch
If your Mac says a software update is required to connect to iPhone or iPad – Apple Support (UK)
If so, you may need to update the software on your Mac in order for it to recognise your iOS device.
Click Install to download and install the software so your Mac can update or sync your iOS device.
It is just a small patch for macOS.
You won't notice that update by the way, it only lasted 3 minutes or so.
The Mac remains Monterey 12.6 and the iPhone remains iOS 16.1.1.