Can the mount path be changed to "/mnt" since this is a system only used by a single person.
Edit, seems this is using udisks2
From the Filesystem Hierarchy Standard:
"This directory is provided so that the system administrator may temporarily mount a filesystem as needed. The content of this directory is a local issue and should not affect the manner in which any program is run.
This directory must not be used by installation programs: a suitable temporary directory not in use by the system must be used instead."
That temporary directory is /run/media/.
You could try a bind mount the auto-mounted folders to the /mnt folder eg:
sudo sh -c "for dir in $(ls /media/$USER | grep -v floppy); do mkdir -p /mnt/$dir; mount --bind /media/$USER/$dir /mnt/$dir; done"
Or create a symlink:
ln -s /media/$USER/drivelabel /mnt/drivelabel