We live in Manchester Twp. NJ and when my Aunt died the nursing home called the wrong funeral home and we had to pay for it, Why?

You should not have paid for anything.  No one other then the family of the deceased has the right to authorize someone to be picked up by a funeral home.  The funeral home should have received verbal permisssion from the family first.  If no family was around and the facility does not have means to hold a deceased, then if a funeral home did them a favor, once the actual family chose their funeral home the other one has to release the deceased without charging anything.  Funeral homes are required to receive verbal consent from family etc. in order to charge a fee.