Initially used as the name for the credit card manager in Windows 98, Microsoft Wallet 'evolved' into the Wallet app included in Windows Phone 8, then regaining its full name in Windows 10 Mobile. Now, the service is once again being renamed: out goes Microsoft Wallet, in comes Microsoft Pay.
While support for the Payment Request API that's in use here was introduced in the Creators Update and could be used via Edge, according to Aggiornamenti Lumia, the rebrand of the service and application pegs minimum installation requirements to Build 16299 or version 1709, better known as the Fall Creators Update. The restriction to Edge and naturally websites that support the Payment Request API is still in place in this new app.
Do keep in mind that as with its predecessor, the Tap to Pay functionality is restricted to the U.S. market at the time of writing, but online payments only require a Microsoft Account, Edge, and a website which supports the Payment Request API.
Source and image: Aggiornamenti Lumia
Thanks to mark in the forums for the tip!