Before you do anything I'm curious if you can drag and drop files located in your Downloads folder, as this is one particular directory usually available for flatpaks by default.
If files from this location work, but not from any other, then you have a good indication that flatpak permissions is the problem. If it doesn't, well, it could still be a permission issue but it simply doesn't allow access to the Downloads directory either. But it's a very quick and test, and could help with troubleshooting anyway.
Regardless of the result you could try to use Flatseal. It's a program also available in the Software Store that it's used to manage permissions to other flatpaks in your system. As I mentioned this "flatpak" package format is designed to run in a sandbox environment which sometimes has this type of weird behaviors.
I have never used it so I'm not sure what advice to give you but it shouldn't be too difficult to find the exact permission that is causing the problem. Note that this is just a guess, after all, so this may not end up solving your issue but it's worth a try. In that case, I would definitely recommend following the instructions from Mozilla to install it manually yourself.
EDIT: Just installed quickly Flatseal and Thunderbird and I think this is the setting that's responsible for allowing access to particular directories. You can specify them yourself or you can grant access to your entire home folder. I guess this warrants a mandatory "proceed at your own risk" just in case, not meant to scare you off but to caution you.
