I know several variants of that tale, Mélusine, including the one you cite. In all the tales she is one of the fey, often a water-spirit, a nymph or a lutine, or something like a mermaid, or with the tail of a fish or a snake. She turns up in some of the Arthurian cycle tales, too. Like the lake-fey of Wales, the French river or fountain nymphs would sometimes marry a human man, or take him as a lover. They would always bring something to the marriage, such as wealth, a herd of faery cattle, secret knowledge or an art or craft or healing spell, which might or might not go with them when they left. Likewise for any children of the union.
One of the Welsh lake-fey produced children who were healers, and even now there is a family claiming descent from her who have a large number of talented doctors among them.
There was always some condition to the marriage, too. For instance, it might be annulled if the husband struck her three times, or said three harsh words to her, or let her handle some jewel or artifact she had brought with her.... The Australian rivergirls will go back to the water if they hear the singing of their sisters, the selchie of Ireland, the Out Isles and Scotland must go back to the sea if they find their seal-skins, and so on.