Does she have to obey her husband’s mother?

With regard to the rights of your husband’s mother and sisters, they have the right to be treated kindly, you should keep in touch with them and treat them kindly as much as possible. 

But with regard to your mother-in-law’s claims that you have to seek her approval in all matters, this is not correct, and none of the scholars has said that this is one of the rights that a husband has over his wife. Rather your duties are those which have been mentioned (by the scholars), namely obeying your husband and seeking his approval, so long as he does not tell you to commit a sin. That does not mean that you should not benefit from your mother-in-law’s experience or her advice if it is useful. And if you are patient in putting up with some of her difficult behaviour in order to honour your husband, that is a good deed for which you will be rewarded in sha Allah. But as for her saying that your family have no rights over you, this is not correct, rather they still have the rights of upholding ties with them, honouring them, treating them kindly and visiting them from time to time, especially your parents. Their rights come after your husband’s rights over you. 

We ask Allah to open your hearts to one another and to grant you wisdom.