Once you have made your final monthly payment and the car credit is fully repaid, you simply have no more outstanding debt with the bank in connection to your car.
Since private credit for car financing is not a leasing contract, there are no additional fees or return conditions—you simply keep the car.
Nothing changes in terms of ownership - you have had the car in your ownership since the beginning of the credit period, and continue to own it after the credit has been paid off. This is standard procedure with car credits in Switzerland.