“Has begun” as in recent? BankAxept has been in use in Norway since 1991 to avoid the VISA/Mastercard “tax” https://en.wikipedia.org/wiki/BankAxept
Biggest banks here refused to support Apple Pay and worked hard on legislations to open NFC access. Now we can pay with no fees to Visa/Mastercard or Apple even from our phones.
There's implementations for this in most European countries (iDeal/Wero, Bizum, MultiBanco, BanContact have been mentioned). What's missing is a unified standard that works across all European webshops and banks.