The move from Android Pay, Google Wallet, and Pay with Google to Google Pay hasn't been completely smooth sailing, but the Mountain View giant is slowly getting its footing and transitioning everything from the old brandings to the new one. The latest to make the switch are web payments done either on desktop or on iOS.
Google is starting to roll out Pay on the web for iPhone, iPad, and desktop users so you should start seeing it when you're trying to make a payment on a supported site, regardless of the browser or device you're using. Your saved cards will also be easily accessible from all platforms so you don't need to add them separately. With this move, Google is slowly closing the gap with Apple Pay and creating a unified front for payments through the Google Pay brand.