Removing duplicate photos from Elements 2019 Catalog

"I thought that elements 2109 would just uses one set of photos from the
files backed up. Unfortunately it has stored both the older back-up versions
of some photos plus newer versions of the same photos from the later backup."

If you did not use the organizer to backup and restore, Elements has stored no file since it only stores links to the files you recovered externally on your drive. If you 'imported' all files from the drive into a new catalog, the import process scanned all your files and ignored true duplicates (files with the same date_taken and size in KB). The file name is not a distinctive criteria. Files with visual similarity are not filtered at that stage, but the organizer has a tool to find them after the import.

That's not very practical.

http://www.johnrellis.com/psedbtool/photoshop-elements-faq.htm#_Find_and_delete

There are external tools to deal with duplicates, but the difficulty is to select the good version and delete the bad ones. The import process in the organizer only keeps the first found file, and rejects the following duplicates without taking account of the location in the folders tree.

Instead of deleting duplicates immediately, it may be interesting to stack the suggested dupes. That does not recover diskspace, but that makes visual searches much easier. You only select the keeper when you need to use it, one by one. That's not the expected magical trick but that works and makes your library progressively cleaner.

Which kind of backup?

A backup from the Organizer?

The backup folder does not contain duplicates. The restore process can create duplicates if you have already copied your files to the new computer from another backup solution.

To solve your problem, you need to explain exactly what you did.