I'm not a finance wonk so I don't know for sure, but given how absolutely everything else in the EU works I'm fairly confident that every country has their own

I'm not a finance wonk so I don't know for sure, but given how absolutely everything else in the EU works I'm fairly confident that every country has their own sovereign financial system for handling currency and then cross-border transfers are facilitated via hacks. It would therefore follow that any international financial system would be very complicated and expensive to set up, and would be helped enormously by a burn-it-to-the-ground-and-start-again replacement system. (...if that system was well designed, and given how everything in the EU works...)