Is zakat obligatory on money a father gave his married daughter to furnish her apartment, knowing that the father is paying installments from it for an investment apartment for his daughter, and who is the zakat payer in this case: the daughter or the father?

The owner of the wealth is the one upon whom zakat becomes obligatory if its conditions are met. If you have acquired the wealth registered in your name and you are able to dispose of it, then this is a complete gift and the wealth has become your property. Zakat becomes obligatory upon you for it if it reaches the nisab and a full year has passed over it. However, if you are unable to dispose of it, then the gift is not complete, and the wealth remains your father's property, and zakat becomes obligatory upon him.