I always strap my cats in when I take them anywhere in the car. They travel inside a blanket-lined kennel/carrier, and the seat/shoulder belts fit nice and snug around it.
We have a kennel for each cat, leftovers from our move across country when we shipped them from California to Virginia.
There are a lot of picky stupid rules about shipping cats that reveal how ignorant the bureaucrats who write those rules are about cats. Stinkin’ idjits.