To be honest, there is already zero way to distinguish between shipping and customer service messages from Amazon. If you order any appreciable amount of items you would have no idea they sent you any message.

Of course I only found this out after being burned by it. Turns out they’d sent me a message telling me the item I returned was not in the same condition it was sent in (it was), but the message was utterly lost in the flood of ‘order received/sent/delivered’ mails they send (with the same subject).

I was about to comment the same thing. It's very simple but I don't think I would have thought of it

Another common technique is a filter to forward all mail. Hard to notice. I ought to go check right now...