Does buying a car from a family member then putting the car in your name and registering it in your name count as a registration fee?

If you paid a registration fee then you paid the fee no matter whom you bought it from. 

General registration fees are not tax-deductible. The only part of the fee that is tax deductible is the part that is considered to be a personal property tax based on the value of the vehicle.  Not all states charge this tax. And in the states that do charge this tax, you must separate your general registration fee from the value tax portion of the fee.

You can also deduct sales tax on a motor vehicle, if you are in a state where you choose to deduct state sales tax instead of state income tax.  Most states will charge sales tax when you register a vehicle, unless the vehicle was a gift from a relative or you can prove that you already paid the sales tax in another state.

personal property tax and sales tax are listed in two different places in TurboTax and are deducted in two different places on your tax return. Make sure you allocate your expenses correctly if you paid either a personal property tax or a general sales tax or both.