Macintosh HD and Macintosh HD - Data are two volumes that Catalina creates and are both valid names and drives. And Catalina needs both. The Data drive has all of the user information and the other has all of the system information. Finder and other parts of Catalina merge these two into one virtual drive for the user so that the user experience is what it has been for a long time, that is, there is a Macintosh HD that represents the boot drive.
However, outside Finder, using Disk Utility, for example, you can see both volumes in the Container. And if you use Finder to look at the backup drive directly, you can see the two volumes separately backed up because TM must do that to recreate the two volumes if you need to restore.
Given that background, how are you getting the "list" of the drives that shows the two? It is perfectly normal and expected to have two.
I suspect the issue is not with TM, but with whatever is triggering the "Backup not ejected properly" message. If that occurs in the middle of a backup, TM will error out. When I have seen that message before has been when I have a drive that has a sleep function, or a power save function. When the drive thinks it's not in use, it powers down. Then when TM wants the drive, the drive has to wake up, spin up and respond, but takes too long to accomplish that before the OS says the drive isn't there, triggers TM to fail and the message about being ejected improperly. About the only cure is to get a new drive, one that either doesn't have the power save function or sleep function, or to use something like Amphetamine (search for it) to query all attached drives every few seconds to keep them spinning.