Glad to see it's not only me. The fact that it's not happening to everyone on Monterey is good to know, but only part of the picture. It may still turn out to have another cause other than macOS Monterey, but I think enough evidence has been presented to not confidently dismiss it as having nothing to do with macOS.
I've been using an M1 Mac Mini running Big Sur for at least six months without any network issues. Within the last several days since upgrading to Monterey, I've suddenly had two or three incidents of inability to reach Internet resources due to DNS errors. And these are sites I've used hundreds of times.
After a few hours of trouble-free operations, my default browser (Edge) will suddenly be unable to resolve any DNS names. I then test multiple websites and browsers (Safari and Chrome) and other Internet connected apps—all with the same DNS failures. Following this, I even tried flushing DNS cache and multiple alternative DNS providers all to no avail. The only thing that has instantly resolved the issue for me (at least for quite a few hours) is restarting macOS.
Several others have already chimed in here with the same problem, so it seems worthy of further investigation.