time Machine fails after Monterey re-install

On a 2015 iMac running Monterey12.7.6, after re-installing OS and then updating last night, I cannot run Time Machine backup to the same HDD I've used for years. It is not full. But somehow, Time Machine thinks I am trying to backup two disks with same name, Macintosh HD. During the recent OS re-install, I copied some files to the refreshed system, but then ran Migration Assistant, which restored a complete backup from the external HDD I am now trying to use. Backups appear on the external HDD going back two years, but I cannot reach them using Time Machine, I can only scroll back about two days.

How to get Time Machine working again?