Is the salary paid by the Swedish government to the questioner and her sister for the care of their mother—given that they both work and have their own income, and their mother also has her own salary—a right for them, or should it be given to the mother?

It is not permissible to take a wage for serving one's parents, because it is an obligation upon the children. However, what is mentioned in the question is not a wage, but rather a reward granted by the awarding entity. This is a right for you and your sister if the government appointed it for both of you. Parents have the right to take from their child's wealth with kindness when there is a need. Satisfying the mother and acceding to her wishes is better for the child and brings greater reward.