Is it permissible for the wife to take the housing allowance for herself without sharing it in the house, especially since she requested it from her work as a housing allowance?

The basic principle is that a mature woman has the freedom to dispose of her own money without her husband's permission or knowledge, whether the money is from her work or otherwise. She is not obligated to spend on the household except as a voluntary contribution, or if her husband stipulated that condition in exchange for her work. As for her utilization of the housing allowance for purposes other than housing rent, it depends on her employment contract. If the employer permits its disposal as she wishes, then there is no harm. Otherwise, she must adhere to the employer's condition.