I think the specific "product" they are referring to is Amazon's willingness to accept the authorization at a certain point and then only actually charge the card later when Kickstarter signaled that the project was funded.
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.
It is specifically this product that Amazon is retiring later this year:
https://payments.amazon.com/help/201626250
They still have other payment products, but I couldn't tell you what the differences are.
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.