It depends a lot on the country, as most European countries have their own established credit card processing firms. These local providers have various advantages in their respective core markets, such as providing service in the local language, supporting local cards and other payment methods that may not be widely used in other regions, easy compliance with local regulations and customer trust through local brand recognition.
For example, Luottokunta (the Finnish Credit Union) has a nearly 100% market share here in Finland. They do offer services outside Finland too, but even there they seem to be mostly focusing on the Nordic and Baltic regions.
I'm sure there are such local providers in Poland too. A quick Google search for credit card processing poland turned up at least eCard S.A., although I suspect I might have got more definitive results if I'd done the search in Polish.