PSE 2020 - Tags

You should restore the backup catalog to the new drive.  That is the only way you can be sure to transfer all of your photos with their tags and other metadata.  It is likely that the reason you cannot see all of the tags from the old computer catalog is that the tags were not saved in the files' metadata.  Tags are only saved if you have run the command in the Organizer to save the information - keyboard shortcut Ctrl+W in Windows, or if the file has been edited and saved from the Editor, after the tag has been applied.

If you did in fact save the tag metadata to the files and imported the tags when you imported the files, then the possibility exists that People, Places and Events tags have been converted to ordinary keyword tags and they may be available under the Imported Keyword Tags category in the Tags panel.  However, you cannot simply drag an Imported Keyword Tag to the People category.  You would have to tag them with a newly created People tag.  And all face recognition information will be lost if you import your photos rather than restore a backup as I recommend.