External drive used for time machine backup not recognized if the drive is moved to a different external drive dock

I have an external SATA drive plugged into a Sabrent drive dock (USB) and it's working fine with Time Machine on my iMac to store backups of the system.

I was just testing the scenario where the drive dock would fail and I would need to plug this drive into a different dock. I have a WD MyBook (USB) so I took the SATA drive out of that and plugged in the hard drive from the dock. My iMac (running the latest version of MacOS, Catalina), pops up the message that it cannot recognize the drive when plugged into the WD MyBook SATA controller and offers to initialize it. Similarly, I did the reverse and took my previously used Time Machine Backup disk from the WD MyBook, plugged it into the Sabrent dock and the iMac cannot recognize that one either. When i put either disk back into the controller that was used to start the time machine backups with that disk, the iMac recognizes the drives with no issues and continues to backup using TimeMachine.

Shouldn't I be able to plug my TimeMachine backup drives into any valid dock/controller and have my iMac recognize it? As long as my current disk dock doesn't fail, I don't need to do this, but want to be prepared for that possibility.

Thanks for the help!

iMac (Late 2015, macOS Catalina 10.15.5)