I don't think pain is an instinct. It's an involuntary chemical reaction between the brain and the nerves to warn you that something is wrong.
^ And you're right V.V.V.V., there are different types of pain. Emotional pain is experienced in animals as well as humans. There are many stories of mother cows who've given birth in factory farms, and have their new borns taken away from them they cry and cry and make so much noise and are without a doubt incredibly distressed that their baby has been taken from them. Sure it's an instinctual reaction, as much of one that any new mother (human or not) would have, but it's also extremely emotional.