The application in question is FreeFileSync, an open-source program (that I definitely recommend) for syncing directories. I use FFS to sync mirror certain directories in my home partition to an external drive. For each such directory, I have a FFS configuration, created in the FFS GUI and stored in a separate XML file (with extension .ffs_gui).
Unlike applications installed from .deb files (where the installer handles the MIME-type stuff for you), FFS comes as an archive that you simply extract and park somewhere, so it does not create its own MIME-type or associate itself with an existing one. On my PC, Mint (with the MATE desktop environment) associated the extension .ffs_gui with XML files in general, so making FFS the default program for .ffs_gui files would have made it the default for all XML files, which is clearly undesirable. So I decided to create a MIME type for it. I also decided to store the necessary information in my home partition, rather than changing system files, so that a future system upgrade would not clobber the changes.
Google searches produced a variety of (rather complicated) solutions, one or two of which I tried but failed to get to work. Fortunately, the steps on the GNOME page mostly did work. First, I created ~/.local/share/mime/packages/application-x-freefilesync.xml as instructed, naming the new MIME type application/x-freefilesync and setting the glob pattern to *.ffs_gui. Second, I created a new desktop file (~/.local/share/applications/freefilesync.desktop) using the new MIME type and pointing to the FFS executable. Finally, I ran
(as myself, not as root).
The GNOME page suggests using the gio program to verify the results, but I don't have that installed on my system. What did work was to open a terminal in the directory where the FFS configuration files were stored, pick one (say, foo.ffs_gui) and run
to query its MIME-type, confirming that the result was application/x-freefilesync.
After that, I right-clicked foo.ffs_gui in the file manager (Caja, on my system), chose Properties > Open With, added FFS and made it the default for all .ffs_gui files, and that was that.