"Subscribe with" implies that you are subscribing to someone else through Amazon, so there's a third party. If you "subscribe to" something, whatever you're subscribing to is the only other party.
With this in mind, I don't think the prepositions are valid in the second pair of sentences. ("Subscribe to Amazon" actually sounds okay to me, but only because I read it as "subscribe to Amazon Prime", which I don't think is what you mean).