The definition of a monoidal structure on $\textbf{Set}$ takes as product structure the direct (categorical) product of sets $X \times Y$.
But this product is only defined up to unique isomorphism - so what exactly do we mean by "the product"? do we choose a representative of the categorical product in the background for each pair of objects $X$, $Y$ and roll with that, or is there something more going on?
(What I mean by something more is that for any two objects $X$ and $Y$ in $\textbf{Set}$, the possible objects $Z$ being a limit of the usual diagram defining the product form a contractible groupoid inside of $\textbf{Set}$. So maps into/out of a representative automatically define maps into/out of any of the other representatives, so everything is "well-defined" in this sense - but is this what's actually going on?)
To have an actual product functor, you have to have chosen products. Any choice will do of course -- we just want any old right adjoint to the diagonal $\textbf{Set} \to \textbf{Set} \times \textbf{Set}$, and any such right adjoint is uniquely isomorphic to any other, as you say.
This can be an annoyance, or even a real hindrance in certain contexts where you want to generalize and do category theory internally inside a category, but there are workarounds. One method uses the notion of anafunctor: see section 3, "Examples", for a discussion of the case of cartesian products.
An old-fashioned but still widely accepted solution is to take "sets" to be ZF-sets and define cartesian products of sets as consisting of ordered pairs of elements a la Kuratowski, $(a, b) = \{\{a\}, \{a, b\}\}$. This is probably the tack that Mac Lane would have taken, had he been pressed on the issue.