Does Apple Pay offer consumer privacy advantages over traditional payment on the web?
Apple Support Document, Apple Pay security and privacy overview, throws some light on the security and privacy offerings when using Apple Pay on Web.

Relevant quote from the article:


  When you use Apple Pay within apps or on the web
  
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  To securely transmit your payment information when you pay in apps or on the web, Apple Pay receives your encrypted transaction and re-encrypts it with a developer-specific key before the transaction information is sent to the developer or payment processor. This key helps ensure that only the app or the website that you’re purchasing from can access your encrypted payment information. Websites must verify their domain every time they offer Apple Pay as a payment option. Like with in-store payments, Apple sends your Device Account Number to the app or website along with the transaction-specific dynamic security code. Neither Apple nor your device sends your actual payment card number to the app.
  
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  When you use Apple Pay on your iPhone or Apple Watch to confirm a purchase from your Mac in Safari, your Mac and the authorizing device communicate over an encrypted channel via Apple servers. Apple doesn’t retain any of this information in a form that personally identifies you. You can disable the ability to use Apple Pay on your Mac in Settings on your iPhone. Go to Wallet &amp; Apple Pay and turn off Allow Payments On Mac.