Mounting an Exfat USB drive

FAT/FAT32/exFAT filesystems do not support users per file as common Linux filesystems, AFAIK. So, chown and chgrp will not work. However you can specify a single user/group for all files during mount

sudo mount /dev/sdc /mnt/media/ -o uid=1000,gid=1000

The group does not have write permissions. To change that add umask=002 to the options in your fstab-line:

UUID=XXXX-XXX /media/my-drive exfat defaults,uid=1000,gid=1002,umask=0002 0 0

The above invocation is for mounting a USB drive into a Samba share.

more about [umask ](permissions - "dmask" and "fmask" mount options - Ask Ubuntu)here.