There are multiple ways to infer that $S_3×S_3$ cannot be isomorphous to the Galois group of an irreducible quintic polynomial.
First as noted elsewhere the order of this group ($36$) does not divide the order of $S_5 (120)$. We might also say that $S_3×S_3$ has two threefold symmetry elements independent of other symmetries whereas $S_5$ has only one, therefore by contradiction $S_3×S_3\not\in S_5$. These two interpretations are of course r wr lated; the second threefold element of $S_3×S_3$ which $S_5$ doesn't have causes the invisibility requirement to fail. If we had given up the second threefold element, specifying $S_3×Z_2$ instead of $S_3×S_3$, we would have met both the divisibility and containment requirements; $S_3×Z_2$ is a possible isomorphism for quintic polynomials.
Note that the last sentence does not contain the word "ireeducible". In an irreducible polynomial of prime degree $p$, the Galois group must contain a $p$-fold symmetry element, and $S_3×S_3$ fails this requirement for $p=5$. In fact even $S_3×Z_2$ fails also. In the latter case, this does not stop $S_3×Z_2$ from being isomorphous to the Galois group of quintic polynomials, but it does mean that such quintic polynomials must be reducible.