I think they can be either and it depends on the context in the same way apps/websites can be features or products.
(1) Feature: a bot that extends an existing product’s value prop into messaging use-cases. (ie. If Amazon made a bot to manage EC2 instances).
(2) Product: standalone bot that delivers it’s own independent value prop (ex. Birdly, Howdy, Meekan, Meya, …)
Disclosure: I’m a founder of Meya, and she exists both as a standalone product, and as a feature when some of her capabilities are licensed for third-party use.