The files shouldn't be owned by root if you mounted the file system using the Disks tool or Caja.
FAT and NTFS filesystems should not have any real file permissions, and if you want to change the owner of the files on such a filesystem, you need to specify that user at the time the filesystem is mounted. But Disks and Caja specify that for you.
This sounds similar to I want write access to ntfs mounted drives. Unfortunately, we never got a reply to the original issue. What I do know is that when I tried mounting an NTFS and a FAT filesystem on a Debian Buster system just a few minutes ago, the files on both filesystems were owned by the user that mounted the filesystem. (Debian Buster has very similar packages to Ubuntu MATE 18.04, and I had Debian Buster handy at the time.)
How did you mount this filesystem?
Show us the contents of /etc/fstab (you can remove any irrelevant entries from the file if you don't want us to see them).
Finally, run the mount -l command as a normal user so we can see who actually mounted the Shared data partition.