I have traveled to The Netherlands, Spain, and Italy with my service dog. Spain and Italy were on a cruise. For the most part I had zero issues with bringing her anywhere. There were a few things that we didn't bother trying to take her in (like the Vatican museum). I was able to take her in to see the Coliseum and she went with me to see The Pope speak. Basically I was asked to show proof of a service dog, which they seemed happy with a letter that I had from my doctor saying that I require her. We also did get a EU Pet Passport to make it easier to between EU Countries.
Also note that while some countries do not recognize service dogs some companies still go by American rules, such as Airbnb and Uber. Really the only person we had a problem with was in Amsterdam an Uber driver refused us but wouldn't cancel the trip (I think if we cancel they still get paid and/or we still get charged). He FINALLY cancelled and we contacted Uber. The rest of the time I have zero problems with Uber.
If anyone has questions about The Netherlands, Spain, and Italy I can try my best to answer! We are taking another cruise next year that leaves from Spain as well.