Statistics is not part of science to the exact same extent that maths is not part of science: by a technicality at most. You also have a very narrow view of what a statistician is if you think they don't spend most of their time applying methods on real data.
I would say that being a "data scientist" is quite different from being a "scientist", with "data science" being more a kind of engineering than a science. For example, the goals of the engineer and the data scientist coincide in the sense that both seek to build products for business purposes. Further, their methods are of a similar kind as in their work towards the product they use statistical models created in house or the industry at large that exist in conceptual frameworks from academia.
