Does the background image come back when you plug the drive into the original Mac? Where is the image stored? On the USB drive or your local hard disk?
Are you correctly unmounting the drive by dragging it to the Trash (which will turn into an eject symbol) before unplugging it? Not doing so can result in data not being correctly written to the drive.
Is the Desktop Services Store being saved correctly? This is a hidden file called
.DS_Store that is saved in every directory and contains information about icon location and background images. To check, open /Applications/Utilities/Terminal, then type
ls -a [drag your disk on to the Terminal window]. This should be created automatically, however there are utilities, such as CleanMyMac that suppress or delete them on external drives and network shares - do you have one installed?
What file system is the drive using? To find out, select it in the Finder, then go File -> Get Info and look at the "Format" field. I'm not sure it'll work if the format is anything other than Mac OS Extended or Mac OS Extended (Journaled) (FAT32 is a common one for USB drives out of the box). It may help to re-format it as such using Disk Utility, although don't do so if you want to use it with a non-Macintosh computer.