What paint for a tire swing?
You could try the type of tire dressing like auto detailers use to spruce up the sidewalls of the tires on cars/trucks. You can find this online by searching for "tire dressing" or "tire shine". It should also be available at your local auto parts store.

My experience with products of this type is that they give the rubber of the tire a surface that is glossy look and a smooth feel. Exposure to sun and weather will likely require re-application from time to time but should certainly reduce the amount of "rub off" coming off the tire.