Is the wife's demand that her husband fulfill his promise to register the apartment in her name considered Islamically forbidden, especially since he has not officially divorced his first wife as he promised?

The husband's promise to buy his wife an apartment before marriage is not legally binding on him according to Sharia (definition); rather, it is recommended to fulfill it. There is nothing forbidden in him buying an apartment and gifting it to her, because he is not obliged to be fair between wives in gifts, presents, dowry, housing, and furniture. Rather, he is obligated to be fair in overnight stays and maintenance according to what is customary. He may prefer one of them over the other in maintenance, desires, and clothing, as long as the other is sufficiently provided for. Similarly, singling out one child with a gift over another is not considered injustice, as equality in gifting is only obligatory for children, not for other relatives.