One of the European Union’s most successful mobile-payment companies is starting its international expansion with an ambitious goal to revolutionize cardless transactions across the bloc.  

A venture of six Polish banks and MasterCard Inc., Blik boasts of 13 million monthly users — which it says is more than any European peer. It is monetizing transfers by taking a slice of e-commerce transactions, grabbing a 70% share of such deals on its home soil last year.