I tried the solution that dibl suggested, it's working very well and is so simple that I feel stupid not finding it out myself. I (as root) made a directory named USERS on the usb-disk and made it write and read able for everybody. No need to hassle around with some complicated (and troublesome) solutions for my needs.
One reason that I didn't found out that solution myself is that I'm to bad in understanding how file-rights work. The directory is marked as read and write able for everybody and that will now be so on every linux system I use the usb-disk on, I had some "weird" thought that the file-rights was connected to one computer or installation, but it's not.
One question, if I do a directory on the usb-disk with only rights for a specific user, will it work for a user with the same name on another computer or installation then?