If you have two accounts, you need two registration codes. One code is good for one and only one account, but you only ever need to use it once. Play that account on another computer, or on a console instead, no problem.
You do, however, need the *software* for each expansion on its respective system. So if you play on PC and 360, you'll need the PC software and the 360 software. But you won't need both of the *codes* that come with those, unless you have two accounts as well.