Earlier quoted context omitted.
The Amazon branded credit card is a MasterCard in the UK. So Visa raises its rates to match MC, but AMZ has a special relationship with the latter.
Note that this special relationship only likely applies to transactions that use the Amazon MasterCard on Amazon.com, and MasterCards issued by other companies still impose full rates. Costco ran into this with American Express, though in retrospect the move to Visa seems like a worse move (more non-Costco Visa cards in circulation).
Could you expand on it a worse move for Costco? I thought it was great they moved to Visa because majority of Americans does not have Amex for credit card. The only way we can use our Visa card is through debit card only. So by moving to Visa (they have largest cardholders than Amex/Discover in America), Costco gained more consumers so the consumers can use their Visa credit card without resorting to debit card. It helps me a lot because I don't and never have Amex. Amex don't have a lot of retailers coverage (they used to be popular back in the 80s/90s), same for Discover. I recalled Amex have higher transaction fees, I believe higher than Discover, and Amex explicitly disallow passing the fee to the consumers. So made sense retailers prefer Visa/MC because they have lower transaction fees. This is what I remember back in the day when Costco made this announcement.