I have a similar issue. Monterey hung on install, and I had to power off. When I fired it up, the installation worked, but my external drive was not seen in the finder or in disk utility.
Eventually, the (parent) drive showed up in DisknUtility, although the child drive, "Time Machine," was greyed out. I ran First Aid, and it seemed okay, but an attempt to erase and reformat failed. Here's a screeshot of what I encountered:
I talked this through with Apple Suypport, and they claimed it was pure coincidence that a drive that had worked perfectly for a year failed immediately after installing Monterey. I've also talked with Western Digital (the drive was a G-DRIVE mobile USB-C) who said they'd had issues like this, but with Macs that were running the M1 chip.
I think this is extremely odd, but am not a wizard with computers. The Mac is a late 2020 iMac i5 3.3Ghz with 32G RAM. Any input gratefully accepted!
Maggot