If you break down GDP in this way, then tourism is most definitely part of the service sector.
Typically, we say that the service sector consists of those businesses that do not produce and sell something tangible. That is, a service is something that you cannot pick up and bring home after you have paid for it.
One example of service workers would be the workers at Disneyland. They are paid to help you get on to the rides, etc. But they do not actually produce something that you can take home and keep. All they are doing is helping you (hopefully) have fun. Fun is not tangible.
Going to Disneyland is tourism. So is going to a resort in Hawaii. In both cases, you do not get something tangible. You are paying for an experience. Therefore, tourism is part of the service sector.