Is it possible to backup a full Linux System on a USB-Stick and reinstall it on the same device again? I have already looked into Software like Clonezilla, but all of those solutions don't seem to work as standalone on boot
Yes, this is easily possible. You could copy every single file from your system to the USB flash drive, and then copy it over to the partition you desire to have it on.
An easier method, which I recommend, would be to take a tarball of your entire system and then extract it to the partition you desire. The last thing you would need to do is make sure that partition's
vmlinuz is recognized from your boot loader so you can boot into it.