The answer is no. A trivial example is to concentrate the measure in a "periodic orbit", this will give an invariant measure for the shift.
But there are a whole lot of invariant measures (including full support measures which probably are more interesting).
The measure which is a product measure, has though some important features. For example, if you look at its "entropy".
(See K. Sigmund, Generic properties for Axiom A diffeomorphisms, Inventiones Math 11 (1970) for the case of the space X being finite)
