Is it possible to have two (separable) Banach spaces, $X$ and $Y$, that are not isometrically isomorphic, and yet their dual spaces $X^*$ and $Y^*$ are isometrically isomorphic?
Yes. For example, $\ell_1$, has $c_0$ and $c$ (the space of sequences that have a limit at infinity with the sup norm) as pre-duals. $c$ and $c_0$ are not isometrically isomorphic (see here). There are many other pre-duals of $\ell_1$.
See this paper for some results on when a Banach space has a unique isometric predual.