You're confusing a dowery with the concepts of bride price and dower.
It was paid to the father but it went to the bride. The dowery was then administered by her husband as part of the family assets. He had no say, however, in its ultimate disposal; and legally, the dowry had to be kept separate because it was intended to support the wife and her children if the husband died. There was no life insurance available, and that's essentially what the dowery was.
So the wife was entitled to her dowry at her husband's death. If she died childless, her dowry reverted to her family, first to her father if he was alive, otherwise her brothers. If she had sons, they would share it equally. Her dowry was inheritable only by her own children, not by her husband's other children or by other women.
Many married women often wore their dowery like you would jewelry with the coins attached to chains or cloth as a way to show off how much their husbands had paid for them.