Another family of infinitely many examples: take $Y$ to be a reflexive Banach space which is not a Hilbert space, then $Y\oplus Y^*$ is isometrically isomorphic to its dual, without being a Hilbert space.
If the isomorphism verifies additional properties, then the result is true. Namely, if a Banach space is isometric to its dual, under certain conditions, it is a Hilbert space. See Theorems 2 and 4 in http://arxiv.org/pdf/0907.1813.pdf and reference therein for similar results. An example of this kind of results is the following:
Theorem: Suppose that $X$ is a Banach space and $\phi:X\to X^*$ is an antilinear isomorphism. If, for all $x\in X$, $\phi(x)(x)$ x$ is orthogonal to $x$ (in Birkhoff-James' sense, ) to $ker(\phi(x))$, then $X$ is a Hilbert space.