I just received a USB Shure MV7 today and had the same problem - would not recognise the microphone as an input device for FaceTime.
MacOS is at 12.2.1 on my M1 MacBook Pro and the Shure was updated to the latest firmware (1.2.13) upon plugging it in to the laptop.
I have been able to work around this issue but it took another piece of software to do it. For this, I used SoundSource by Rogue Amoeba software. In the SoundSource interface, I changed the System Input to be "Shure MV7". I then added FaceTime to the Favorites section and chose "Shure MV7".
In FaceTime, I configured it to use "Use System Setting" in Video -> Output. (Note: there is no setting for Input).
This worked without issue. When you unplug the MV7, the sound settings go back to their defaults. Settings for the microphone are remembered, however, and will automatically configure themselves when the mic is plugged back in to the laptop.
Frankly, it's a pain to have to do this and it adds expense but at least my brand new microphone will work with my machine. Really looking forward to Apple fixing this, though, as it seems it is very clearly their problem.