Is part of a car bought by a sister with her mother's money and her salary from her work considered haram if the mother did not pay zakat on the money with which she bought the car? And is it obligatory to sell the car to pay zakat from it if zakat is due on the money itself, and does the zakat multiply as the price of the car multiplies?

The obligatory Zakat of the deceased mother must be disbursed because it is a debt upon her estate, and the debt owed to Allah is more deserving of being fulfilled. As for the car purchased with the money on which Zakat became due, it is your sister's property, and she is permitted to drive it, and there is no need to sell it. This is because the purchase is valid. Although Zakat relates to the asset itself, it primarily relates to the financial liability of the individual (dhimma). Ibn Qudamah stated that disposing of money on which Zakat is due, through sale or gift, is permissible. Ownership of the zakatable amount (nisab) is not established for the poor; rather, Zakat is an obligation upon the individual's financial liability.