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
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.
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.