What would a psychiatrist tell a teen who is obsessed with cats, all kinds of cats?

Cat obsession? This would depend on what the "cat obsession" actually looks like. Is the teenager hoarding cats? Could be a type of obsessional anxiety. Is the teen intent on harming or killing cats? Could be an antisocial personality showing up, or signs of being abused himself or herself. To say anything useful at all, the psychiatrist would first need to hear what's happening directly from the patient.
Agree w Dr. Pappas. I would want to explore the meaning of the cats with the teenager. I would be concerned it the issues with cats interfered with other areas of living (getting physical activity, socializing with others, etc.).