Is the money paid to the mother after childbirth (Naqūṭ) her right or the right of the newborn according to Sharia, and is it forbidden for her to take it?

If the gifts are specifically for the newborn, such as clothes, then they belong to the newborn. As for money and similar items, if custom dictates that they are for the mother, then they are hers. If custom dictates that they are for the newborn, then they are his. The mother has the right to dispose of what belongs to her.