Yes, you can back up as many computers as you want on a single My Passport. It will automatically recognize the correct computer when you plug it in.
You can also use up to 3 Passports on a sigle computer. I do this so one is backing up, one is safe at home, and one is in transit between work and home or home and work. That way I always have a fairly recent backup at home should the worst happen.
And since you can back up multible computers per Passport, why not back up the home computer with the drive sitting at home. Now you have both home and work backed up and 3 Passports cycling around. Should the worst happen to your house, your home files are safe at work.