Time Machine Glitch

Back up your Mac with Time Machine - Apple Support

The +/- on the first Time Machine window is for adding or removing a drive for Time Machine to backup to, it is not for adding another external drive to get backed up.

To include an external ssd for inclusion in the TM backup routine, simply connect it to your Mac, and in the TM window click the Options button and make sure the ssd is NOT on the exclusion list.

Make sure your TM backup drive has enough capacity to backup all the drives. Ideally the backup drive should be 2x-3x the total capacity of all the drive you wish to backup. So, to backup three 500GB drives - 1.5TB total - you'll want a TM backup drive with a capacity of 3TB or more.

Okay, please try this.

1. Please shut down the computer and physically disconnect the 2 external drives.
2. Then restart the computer in Safe Mode , if you use a wireless keyboard please plug in the charging cable that it came with.
3. Next, restart normally and connect each of the external drives and see if they are seen by Disk Utility (Applications - Utilities - Disk Utility)
4. Ensure both external drives are formatted to APFS as noted in Disk Utility.
5. Then set them to be backed up by Time Machine or not as you desire.
6. Then tomorrow after Time Machine has done its job, test it as I described earlier.

If a simple reboot of your Mac doesn’t fix things, see if the problem still happens in Safe mode. It can take much longer to safe boot (5 min) so be patient.

How to use safe mode on your Mac - Apple Support 

Safe mode can often correct weird software behavior by forcing the OS to do cache clearing and other housekeeping. When you're in safe boot, the machine will not be at its best performance, especially with graphics, but that's expected.

Does the same thing happen while in Safe mode?

You can exit safe mode by restarting your Mac normally and re-evaluate the issue again.

Time Machine backs up everything that is connected to the iMac unless you tell it otherwise! If you have drives, files, folders that you DO NOT want to backup then you add them using the +- symbols. To test if Time Machine is working correctly after you have the settings you want, let it run for 1 full day then select a file that you want to restore (I normally use a file I don't really care about) and attempt to restore it. If it restores okay then Time Machine is working fine.

When you reformat that will wipe them clean so if you have info o them you want to save you need to save that onto a 3rd drive or some other storage device.