Inheriting the old Mac's Time Machine history on a replacement Mac

I wanted to share my solution to this issue. After migrating data to a new Mac, when I connected my older Time Machine disk, it prompted me for the password (since it's encrypted), but never prompted if I wanted to re-use the older Time Machine history. Before trying some of the solutions using the tmutil command, I tried one other thing.

At this point the TM was mounted, but backup would fail. I ejected the TM disk, opened up the TM settings which showed a red error icon next to the TM disk. Clicking on it finally offered the choice to "claim" the older Mac's TM history, which I did. After re-mounting the disk and entering the password, the backups worked.

This is on my wife's new Mac Mini M4 running macOS 18.1