Unique up to unique isomorphism

When people say, for example, that the product $X \times Y$ of two objects $X, Y$ is unique up to unique isomorphism, that doesn't mean that $X \times Y$, as an object of the category, has no non-trivial automorphisms; it's easy to find examples where this is blatantly false. It means that if you have two objects $A, B$ which are both products of $X$ and $Y$ in the sense that they come with distinguished projection maps to $X$ and $Y$ satisfying the universal property, then there is a unique isomorphism $A \to B$ compatible with the projection maps. The projection maps are part of the data that defines a product, and in particular it is possible for the same object to be a product of $A$ and $B$ in two different ways (in the sense that the projection maps are different): those different ways are then related by an automorphism of the object.

Another way to say this is to say that a product is a terminal object in a certain category of cones, and as an object of this category, it follows that the product has no non-trivial automorphisms because, for any terminal object $1$, there is a unique map $1 \to 1$, which must be the identity.