I agree, its not what you named her and if your not happy with the nic name (even if she ends up being called that later in life) then say something.
Just pull aside one of the staff who you know will listen and will make sure the message gets passed on, that you have noticed your daughter being called 'ani' and that its not what you call her. Tell them you dont shorten her name and no one outside of preschool does, she always goes by her full name.
If you notice in a week or so her still being called 'ani', then make a complaint of it to the manager.