You can certainly use one desktop to manage both BlackBerry devices. The Desktop Manager works with the PIN number and will differentiate the two.
Does the problem device use a password? If you are not entering the password on DM when connecting the device, this would cause the symptoms you're seeing.
What happens if you DO enable mass storage? Is all well?
Treat your password like your toothbrush. Don't let anybody else use it, and get a new one every six months.