Don’t try it with a cat because they see your head as the most stable platform in the airplane, and under those conditions they find human flesh the best surface to get a good grip on.
Most cat owners have survived the experience of having a loose cat in a car, and don’t ever want to duplicate it, but a pilot might not get a second chance.
Back when I lived in Hollywood I used to bring my roommate's cat along when I would go to pick her up.
It would usually straddle my shoulders behind my neck and watch other drivers.
Even in jaded Hollywood people will stare at a cat in a car...
One of my cats used to go for long rides with my wife and me. She would always want to sit in the lap of whoever was driving. I always considered it a position of dominance, but stability also makes sense.