Each copy of the game comes with a 'key' for the system it's for - that key is attached to the account you register with it. It enables you to use that system's version of XIV under that account. You can have multiple keys registered to a single account (eg an account enabled for PS3 and PC, which can now play on either whenever it wants); however you can't register a single key to multiple accounts.

As mentioned, to do what you want, you'll need two accounts - each with a key for PC and PS3. If that sounds too expensive, have one of you pick each system and stick with that one.