Is the wife correct in her belief that it is more befitting for the husband to provide money for them to purchase their own home, and what is the legitimate Islamic advice for dealing with this situation, and does the wife have the right to demand a fixed monthly allowance for herself to save from for the couple's future?

The mother's right is great, and honoring her is among the best deeds. She should not take from her son's money without his consent if she has sufficient means. If the husband spends on his wife and children in a reasonable manner, then the wife has no right to object to his spending on his mother or gifting her from his money. However, there is no harm in discussing and advising him to balance that with saving money to buy a house.

A wife's demand for a special monthly allowance is not obligatory for the husband.

If the husband takes out a usurious loan to buy a house or for any other purpose, he has committed a grave sin. He is not permitted to obey his mother in this regard because it is an act of disobedience to the Creator, and he must repent from it.