It all depends on the filesystem on the device. If it’s a POSIX filesystem, then the POSIX permissions apply, and then root-owned files will remain root-owned files.
If it’s one of the typical Microsoft-style filesystems, then file ownership and permissions are faked in the virtual filesystem layer, because those are not POSIX-compatible filestystems, and in that case, it should be automatically mounted with yourself as the (fake )owner.
Still, you might want to consider setting up a static /etc/fstab entry and a mountpoint for the filesystem iinside your home directory.
