In addition to the interchange fee, the payment processors and the banks involved in the transaction charge their own fees. When I last checked it in Finland a few years ago, the list price from a traditional payment processor with a monthly fee was a little under 1%, assuming that the card was a European consumer credit card. Business cards and foreign cards could have higher fees, up to around 1.5%. Non-traditional payment processors without a monthly fee charged a bit under 2% for every transaction, regardless of the card.
Also, regulatory caps are not the real reason for lower fees. In some European countries, banks were traditionally focused on efficiency. Because merchants were used to cheap payment methods, credit cards didn't become widely accepted until the fees got low enough. Then the EU and the common market happened. The efficient banks with low fees realized they could gain market share in other countries by offering cheaper payment services. Visa and MasterCard didn't like that and tried to ban the practice. Then the EU didn't like that and eventually ended up imposing a cap on interchange fees.