I use an Amazon.com Rewards Visa® Card for my Kindle purchases, and this works like a standard Visa card but with cash back for Amazon purchases (including, I think, Kindle purchases).
You probably have an Amazon.com Store Card, which is different to the Visa card. Its
FAQ says:
Quote:
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You can use your Amazon.com Store Card for any purchase at Amazon.com — with the exception of digital items, including Amazon MP3s and Amazon Video On Demand video.
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This does not mention Kindle ebooks directly, but they are definitely "digital items". It isn't really clear why the Store Card would not work for Kindle ebooks, since this at a minimum saves Amazon the 3rd party credit card overhead.