

The EU is not a country.
Some countries in the EU currently have or have had Social Democratic governments, but mainly they have governments which are Neoliberal, though a milder form than the US: generally the mainstream Rightwing around this parts has policies which are to the left of the Democrat Party in the US, though not by much, so for example nobody has a Healthcare system which is as bad as the US - even the ones with a Health Insurance based system have way more rules and consumer protections around it - and even in the worst countries Public Transport is better than in the in US.
Then again at least one country in the EU - Hungary - currently has Fascism whilst the other ones which are said to have Far-Right governments (such as Italy) politically sit between the US Democrats and Republicans.
In the things which are the responsibility of the EU (i.e. trade-related subjects), the EU is significantly more pro-consumer than the US, with for example the precautionary principle - i.e. proven safe before allowed, rather than the US’ method of allowing until proven unsafe - being used for chemical substances which people tend to come in contact with, and more broadly with consumers having way more rights all across the EU than they have in the US (were it massively depends on the State) and with stricter rules when it comes to pollution and more broadly Environmental damage.
I supposed that in the things which fall under the responsibility of the EU, it tends to be sort of half-way between Neoliberal and Social-Democrat, for example it’s very Neoliberal when it comes to Finance, but it’s Social Democrat when it comes to consumer rights and protections, especially for things like food, though even there it’s sort of somewhere between lax and strict in regulatory terms. I suspect this is due to different countries caring more about different domains and hence the politics of countries which care more about a specific domain getting more strongly imprinted in legislation at an EU level so it ends up reflected into very different political spins for different trade domains.
I know that both Portugal and The Netherlands also have their own local systems, but you can’t really use the system of one country in another country.
The only country in Europe which I know for sure doesn’t have its own local payments system is the UK, though it would not surprise me if there are others.
What’s really needed is some sort of pan-european payments system, ideally one which also gets accepted in the rest of the World. The closest we have to it at the moment in the EU is that you can do normal (so called SEPA, if I remember it correctly) bank transfers to any account at any bank in the EU, all for the same cost (generally free) independently of it being in the same country or across borders, and quite a number of retailers all over Europe do accept payment via bank transfer, but that’s not an actual payment system, it’s a bank transfer system that you sometimes can use to pay an online order from a retailer.
As things stand now, if for example from my Portuguese bank account I want to buy something from an online store in Germany, the payment has to go via Visa (Mastercard isn’t really common in Portugal)