I think that a merchant can charge a fee if the card interchange fee isn't capped under some EU regulation or directive. The fee charged by the merchant may not exceed the fee the merchant has to pay to the bank, but this fee maybe depends on the merchant's contract with the bank.

You will have to answer these questions, I think:

Was the card issued in the European Economic Area?

Does the card network have the concept of an interchange fee?
Yes: Visa, MasterCard
No: American Express

Is it a personal card?

If the answer is yes to all three questions, then there should not be a fee. If the answer is no to at least one of the questions, then there may be a fee, but the merchant may choose not to charge a fee.

In the past, there were many shops which only accepted Dankort (which requires a Danish bank account), but thankfully those shops have become very uncommon. I always bring some cash in case a restaurant only takes Dankort. It would be embarrassing if I couldn't pay after eating.