$A$ is never isomorphic to $A[x]$ as an $A$-algebra. Because, for any $A$-algebra $B$, the set of $A$-algebras homomorphisms
- $Hom_A(A,B)$ has only one element: the unit map $a\mapsto a\cdot 1$.
- $Hom_A(A[x],B)$ is canonically identified with the set $B$: $b$ corresponds to $\sum a_nx^n\mapsto \sum a_n b^n$.
But it is possible for $A$ and $A[x]$ to be isomorphic as rings. Take $A = k[x_1,\ldots,x_n,\ldots]$ a polynomial ring in a infinity of variables, then $A \to A[x_0]$ that sends $x(i+1)$ to $x_i$ is an isomorphism of $k$-algebras (not of $A$-algebras).