The product/service distinction is orthogonal to the relevant category in (U.S.) copyright law: the work. Only "works" can be copyrighted, but a work can be provided as part of a service, sold as a product, or one of any number of other possibilities. Nice try though. :)
But it's still a product! I can sum this up that all services are products but not all products are services. So if I were to publish a book which would classify as "work" which I can copyright. At this point, I have a product that doesn't exist yet other than a word document ready to be sent off to publishers. The publisher will take my "work" and print it out into a physical book which we can clasify as a product. They can also format it into a electronic version of the book which is also a product. Now, how the publisher can send it off to a service provider like Amazon Kindle to distribute it (that's a service)
Theft of services is a common enough occurrence (e.g. someone sneaking a bag of garbage into the dumpster of a business instead of paying for trash pickup).