The European alternative was sold to--sorry, merged into--Mastercard and rebranded to Maestro: https://en.wikipedia.org/wiki/Europay_International. It was the E in EMV. So friendly were European and American relations in the past that it was allowed to happen.
I don't think there's such a thing yet for payment terminals. Local alternatives all seem to have died off because banks in other countries didn't support them.
I hope the new attempt you linked will make a difference. This time, international support seems to be at the basis of the system. Widespread integration of a unified form of iDeal/Bancontact/Payconiq could easily replace the need for American style credit card forms across Europe.
Technology-wise this is a solved problem. Uses open-source software too. All the EU needs to do is say 'self-custody is fine, please use a EUR-based stable coin for payments EU-wide'. The market will do the rest (the stable-coin, payment terminals, integrations with cash registers, DeFi forex for places where EUR is not used, etc.).
I hope there will be more adoption of EU stable coins like https://monerium.com, and a payments ecosystem will form around it. The rails are there, we need the apps and the integrations next.
