Time Machine problem

I found that you have to Re-Format the external drive with the new laptop's Disk Utility to

APFS (Case Sensitive) even if your external drive was formatted APFS (Case Sensitive) out of the box.

That happened to me and was not getting any backups until I did this. Then it seems to work perfectly.

It cannot be a backup drive nor a startup drive, and it must be formatted APFS or Mac OS Extended.

If all of that is met, you should be able to remove it from the exclusion list in Time Machine Settings.

No, Mac OS Extended is very old.

Reformatting a drive will erase everything on it. The only reason you would need to is if it wasn’t using either of the two macOS file systems.

Is the drive listed in the Exclusion list and cannot be removed?

Correct.

If that is in reference to it not being able to remove it from the exclusion list, you have to figure out what is odd about the drive. Time Machine thinks it is not eligible for backup.

I would copy the files to another drive, erase the drive, and then copy the files back to it. Then, try to back it up.

Is there anything else on the backup drive? You could use that for the copy. Might be safer to use another.

Time machine not backing up an external hard drive.

Does this mean that you have an external hard drive connected to the Mac and you want to back up both the Mac and the external drive to another backup drive using Time Machine?

or

Does this mean that you are trying to back up the Mac to an external hard drive?

most probably, the size is small or just try another one, i suggest a 25 gb one