Product of two objects in category is unique _up to isomorphism_. That is, if you found two objects that satisfy the definition of product, then there exists a _unique isomorphism_ between these two products that respects the product structure (remember that a product is not just an object, but also two projection morphisms). This allows one to speak about _the_ product of objects (since, from category-theoretical point of view, isomorphic objects are indistinguishable). This statement can be easily proven using the defining _universal property_ of products.
Note that products are not guaranteed to _exist_ in a particular category (though a lot of example categories do have products, like the categories of sets, topological spaces, universal algebras of any kind, including monoids, groups, rings, etc).