I don't see how the police could even say it was that family's car unless you could see the license plate or somehow prove it is the only car of that model and year in the entire state with that state's license plate.
The thing to do in that situation is not even tell the police who was driving the car, even if it's your family car. Let them prove who it was beyond a reasonable doubt.
Any 1st year lawyer will tell you to never speak to police unless your lawyer is present but most US citizens still have the idea that speaking with police might help. That is not the way police operate now days.
As to the plate, maybe they had a partial and matched it to a sort of similar car then ran with that instead of actually doing their jobs?
Having watched 1000's of cases now, I am of the opinion that a very significant potion of police misconduct originates with laziness on the part of the officers.