Suppose $S$ to be a Polish space and $P(S \times S)$ the set of probability measures on the space $S \times S$. If $\nu$, $\mu$ are in $P(S)$, we can construct the probability measure $\gamma$ on $S \times S$ by setting $\gamma (A,B) = \nu (A) \mu (B)$ (for sake of simplicity, I am omitting here further details on the probability space, but a nice refreshing is given by Wikipedia https://en.wikipedia.org/wiki/Product_measure). We use to say that the product measure $\gamma$, usually indicated with $\nu \otimes \mu$ has marginals $\nu$ and $\mu$ because of the multiplication property in its definition. Furthermore, since we are in a Polish space such a product measure exists and is unique, consequently $\nu \otimes \mu$ is the unique measure $\eta$ on $S \times S$ satisfying $\eta (A,B) = \nu(A) \mu(B)$.
Suppose now to have a generic $\hat{\gamma} \in P(S \times S)$. What does it mean the statement "$\hat{\gamma}$ has marginals $\nu$ and $\mu$"? Does it just refer to the multiplication as before, i.e. $\hat{\gamma} (A,B) = \nu(A)\mu(B)$ ? If so, thanks to its uniqueness, why isn't $\hat{\gamma}$ the product measure $\nu \otimes \mu$?
I am trying to understand the definition of Wasserstein metric (https://en.wikipedia.org/wiki/Wasserstein_metric), but I find the part "...denotes the collection of all measures on $(M \times M)$ with marginals $\mu$ and $\nu$..." a bit unclear. The hyperlink "marginal" there brings to the page about "marginal distribution of random variables", which is clear for me, instead of "marginal distribution of measures", which is what I would like to understand.
Furthermore, it seems that the set we are dealing with is commonly called the collection of couplings between $\mu$ and $\nu$. Searching again, this page of Wikipedia (https://en.wikipedia.org/wiki/Coupling_%28probability%29) is this time very clear, but it slightly different from the set found before. Here we are considering random variables and their distributions (and so no problem in speaking about marginals), while the definition that generated my question only involved probability measures (and so my trouble in understanding the relation between having marginals and being the product measure).
Any kind of hint/help/suggestion would be very appreciated!
Ps: I am pretty sure that the wrong step in my reasoning should be in deducing "$\hat{\gamma}$ is the product measure $\nu \otimes \mu$" from "$\hat{\gamma}$ has marginals $\nu$ and $\mu$". I am trying by my own to understand why, but a counterexample would be particularly appreciated.