Technically, Amazon isn’t storing your credit card information. Yes, they’re displaying the type of card, the last 4 digits, and the expiration date, but that’s all they keep. What happens is the first time you use your credit card it is sent via a secure link to something called a “middleware” company in the credit card business. They take your information, forwards that little bit that Amazon shows, but then they take the meat of the data, and they create a “token” that they forward to Amazon.
That “token” can only be used by Amazon, and it can only be used by Amazon if you are logged into your Amazon account, and it can only be used if the shipping address is the one that was used on that first purchase. (That’s why you get asked to re-enter your card information if you change the shipping address.)
The middleware company doesn’t keep your real credit card information, either. They don’t store your encrypted token, either. Only Amazon has it.
So the worst thing that someone can do with a credit card you have associated with your Amazon account is hack into your Amazon account, and buy something to be shipped to an address you’ve already used that card to ship to.
Your card information is at a greater risk only once: When you first enter it to make that first purchase. And then the risk is that it may be intercepted before it even hits your browser. Someone could install a “key logger” on your computer, and every time you type a 16-digit number starting with 3, 4, 5 or 6, it sends that number, and whatever you’ve typed just before and after that number, and then transmit it to the bad guys.
So if you want to exponentially raise your risk of having the credit card you use on Amazon being hacked, delete it after every purchase, and then type it in again each time. That’s FAR, FAR, FAR more risky than letting Amazon keep that token.