Is what the sister mentioned, that caring for the mother is the responsibility of the sons and not the daughters, correct, and that the right of the husband and children is more obligatory in Islamic law, and is this what the upright Sharia brought?

If the parents need service, then it is obligatory upon all their children, male and female, to provide it, either by themselves or by hiring someone to perform the service. This is a religious obligation and part of the commanded righteousness. If the husband prevents his wife from serving her mother, then the husband's right takes precedence, and she should compensate for that by hiring a maid or contributing to her wages if she has money. The sister should be advised to show righteousness to her mother, and that a daughter is more suitable to serve her mother than a son, especially if the mother needs someone to attend to her affairs from among women.