I see. I had not used that previously. I installed Backup 1.1.3 by Rob Weber on one of my RPi’s running OSMC and did a uncompressed backup to a SMB share and that backed up exactly how it was suppose to. I then pulled my userdata on a Windows box so I had a fresh empty Kodi. I then installed the Backup add-on, pointed it to my SMB share, and told it to restore. What I was left with at this point, and after restarting Kodi, was a working library, missing add-ons, no profiles, and a no settings which were stored in guisettings.xml (which is a good thing in this case and expected as this is not something you can normally replace with Kodi running. It then dawned on me that the settings in the Backup add-on were for both backup and restore so I checked the rest of the boxes and told it to go again. That got things a bit closer but add-ons were not enabled and profiles were not enabled. So basicly the backup seems fine but the restore has limits. I would expect Kodi on a Fire Stick to act much the same as Windows.
I then dumped my Kodi folder on Windows once more, opened Kodi so it made new default files, and then with Kodi closed I manually moved the files from the Backup replacing anything that was there. After this Kodi was fully the same as my OSMC box. I assume the Backup program has limitations trying to do the restore with Kodi open.
As such I would check your NAS to see if the backup is actually there. If not check the add-on’s settings and try again. Alternatively you could manually copy them instead of using Kodi. Once you have the files then I would use ES File Explorer to copy these files instead of trying to do it in Kodi on the Firestick.