It's entirely dependent on whether the merchant has opted into the product via their payment provider. I don't think it's particularly prevalent here in the UK yet. I'm sure Amazon will have it very soon.
OK, sounds like there was an unfortunate cock up in your particular instance. The new card details should only be sent by the issuer and used by the merchant when you have activated your new card.
I used to work in the industry and this is exactly what happens as described by Goldstone1976. It is meant mainly as a positive customer experience so that you don't have to remember everywhere you have your card details on file, e.g. magazine subscription or insurance services etc. I...