The reason you can't has to do with the rule for PlayOnline accounts that it is suppoed to only ever be one person using an account and they aren't supposed to be transferrable. You weren't supposed to create a character for a family member on your account, they were supposed to get their own account. That should have been obvious from not being able to play both characters on the account at the same time, and you should have gone out and bough another copy of the game when you noticed that, before you had invested months/years of time in the character. SE IDs and security tokens further enforce that rule.

So now you've been leveling multiple characters on the cheap (not paying a full monthly fee per character), and want to be able to use them all at once now?