Hey Bupadin,
I’m not sure I understand correctly here, but from what I can tell you have done the achievement on your character and then faction changed.
You want to be able to do the daily and achievement on another character as well? In that case, you would need to re-do the questline that unlocks it.
If you have done the questline and you do not see the daily, I would suggest that you report it as a potential bug.