It does not work that way in my area at all. The lender is the leinholder and is not on the title as an owner at all, merely as a leinholder. Then the lender releases the lein when the car is paid off, which does not change anything on the title except removing a leinholder. (on a technical basis the leinholder appears on the bottom of the title, in a totally separate area from the owners) Again, in my area the title and the registration are separate things. One can be on the title (and typically all parties to a loan are placed on the title by the car dealer) but not on the registration.Until the vehicle loan is paid in full, the lender will typically be the title holder/owner of the vehicle. The lender’s name will appear on the title along with the car purchaser. If payments are not made as agreed in the loan documents, the lender can repossess the vehicle.
However, even with the car title held by the lender and the lender the legal owner, the dad probably registered the car in his name rather than in sister’s name, and the insurance policy apparently is/was in dad’s name. That would make dad the registered driver and sister a permitted user.
The dad and sister are both responsible for the car loan, since dad co-signed on the loan. But the loan financing does not equal car ownership. Once the car loan is paid off, there will be a release of the lien and application can then be made with the DMV for an updated title.
Car registration can be complicated if there are unpaid vehicle violations, like toll fees and fines.