In the US, you also wave your phone over the card reader, it goes "Pling!" and that's your stuff paid for. The GP comment is not about paying (although it can do that too [1]), but rather about providing their Amazon account details for Amazon Prime discounts and other benefits.

A similar process is the case in the UK as well at Amazon Fresh stores, last I checked.

[1] https://www.amazon.com/gp/help/customer/display.html?nodeId=...