Verify that $S^{-1}A$ satisfies the following universal property: $S^{-1}A$ is initial among $A$-algebras $B$ where every element of $S$ is sent to an invertible element in $B$.
Won't $S^{-1}A[x]$ also be an initial object in this category? Note that $S^{-1}A[x]$ is not isomorphic to $S^{-1}A$.
In all homomorphisms $\phi:S^{-1}A[x]\to O$, where $O$ is any other object in the category, map $x$ to $1_O$.
$S^{-1}A[x]$ is not initial in this category because there is not a unique map $S^{-1}A[x] \to S^{-1}A$, there are many such maps.
On the other hand if $B$ is an $A$ algebra satisfying the conditions then you can show that there is only one $A$-algebra map $S^{-1}A \to B$.