By looking at the source it is a database menu.
So go in admin->menu->application menu options and checked the permission or groups that are on the admin label lines. (there are a couple of lines to give the list of perms allowing the label to be displayed)
Ever your anononymous are explicitly given in a group of one admin line, or your anonymous has a perm that allows the label to be see (the most probable). Compare the list of perms the admin label and the list of perms that your anonymous has (admin->groups->perms). I bet anonymous has a perm they should have and there is a bug in the perms on the menu.