Do you remember MS-DOS ? it's the place where FAT filesystems born. MS-DOS was a single-user system, no owner, no groups, no others.
Accordingly to this "limited" view, FAT filesystems doesn't have any attribute to classifies file and directories ownership.
Unix/Linux, on the other side has always implemented the "*nix classic" ownership/permission scheme owner, group, others.
How can be mounted a FAT filesystem on a Unix Filesystem ?
simulating the ownership/permission model.
Therefore yes, if you mount a fat filesystem as 'gert', the files will appear as owned by 'gert'. This is limited to FAT filesystem family and does not apply
to Linux/Unix native filesystems, therefore the behavior will be different depending on filesystem type.
When you 'gert' copy a file, you will create a new file and this will be owned by 'gert' even if the original/source is owned by some other user,
you 'gert' only need the read permission on that file to copy it, and then you will own the copy.
If I mount any vfat USB stick (have a number of these to exchange files between platforms) and then look at ownership I see following
when you mount a vfat, files will appear as owned from the current user.
[gert@localhost ~]$ ls -l /media
total 36
drwxr-xr-x 27 gert root 16384 May 5 23:57 TOSHIBA/
[gert@localhost ~]$ ls -l /media/TOSHIBA/
total 1376
-rwxr-xr-x 1 gert root 8962 Nov 22 00:11 00-ToDo_Items_Testing.glk*
drwxr-xr-x 2 gert root 32768 Jan 24 10:59 01008xx-PCLOS_printing_duplex_problem/
drwxr-xr-x 2 gert root 32768 Sep 22 2010 100722-sunnyday/
..
..
....
The file on the desktop I can not drag to the USB drive (I could not save it to the USB stick either of course) I guess it is fair enough a user can not save to a root group??
No, as far as I can see you 'gert' as owner of the USB stick contents, should have full read/write access to your USB stick...

let me check about, it sound like something is bugged, I will come back later about.
AS