Normally, Thunderbird creates a hidden folder to store the profile information under your user's home directory (that's just the folder that you see when you launch the file manager, where you have your Documents, Music, etc.). Since the Flatpak version didn't have access to this location before, it probably used another location. Now that you used Flatseal to give it permissions to all files, it sees that there isn't a profile at the expected location, and creates one. That's at least my thinking...
You can fix that in one of two ways:
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Since the goal is to enable drag & drop, at least for the time being, you could just pick and choose one or a few locations that you want to use for that purpose. The Downloads folder is typically already accessible, but you can also add Pictures, Documents, Desktop, etc. This should keep things relatively simple, with only a slight inconvenience in the sense that you need to remember about this.
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If you prefer to add access to all files in your home directory, you can copy your current profile folder to the new expected location. Or, even better, just create a link that will effectively work in the same way but will allow you to backup your configuration much easier.
First you need to delete the new profile directory that was created after you granted access to all of your files.
Close Thunderbird, and open the file manager. On the settings, you can toggle visibility for hidden files, which will reveal a folder called ".thunderbird". Instead of deleting it, move it to your desktop or somewhere else; that will be enough and it's much safer, just in case.
Then, launch a terminal from the applications menu and run the following command:ln -s ~/.var/app/org.mozilla.thunderbird/.thunderbird ~/.thunderbird
This will create a link — a shortcut in Windows lingo — where Thunderbird expects to find a profile.
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Last but not least, you can uninstall the Flatpak version and go with the native Debian package that will work as expected. Since you are already running things I would leave this for last, but know it's an option.