It is good for Amazon because Amazon Payments was not the best service from Amazon, probably had the most non Amazon like experiences within it mainly bad customer service and worse than Paypal fraud false positives. For doing such a good job in all other areas they enter usually, here they made a worse PayPal, exactly the thing they were trying to better.
Stripe is such a better alternative to Amazon (and paypal which Amazon payments was supposed to rival). Amazon payments is now more business to business or person to business and no more person to person payments.
I'm not necessarily saying you're wrong, I'm just not sure what else that could mean. When I see a button to pay with Amazon, the button says "Amazon Payments" on it, which I, as a layman, would call the Amazon Payments product if I had to call it anything.
Of course, Kickstarter may have just misspoke. Or, like you said, it's a specific product that only applied to Kickstarter. Either way, it's quite confusing from where I'm sitting.
I remember hearing a long time ago that Amazon wasn't particularly happy with this arrangement, and had specifically declined to offer it to Kickstarter competitors.
https://payments.amazon.com/help/201626250
They still have other payment products, but I couldn't tell you what the differences are.