I feel closer to all animals, too.
I realize that, unlike me, a cat is an obligate carnivore. What makes me feel close is realizing that like me he is an individual with a will to live, a personality, interests, and fears.
BTW, a cat "plays" with his prey as a self-defense and survival strategy. By dazing and weakening the prey-who has teeth and claws also-before coming in close for the "killing bite," which severs the prey's neck and kills him quickly, the cat decreases his risk of injury or the prey escaping.