I'm studying an example in book Yuji Shimizu and Kenji Ueno. Advances in Moduli Theory. Translations of Mathematical Monographs, vol. 206, that shows the importance of isomorphism as principally polarized abelian varieties in Torelli's theorem.

I'm studying the following example:

For a compact Riemann surface $R$ of genus $2$, $W^1=\varphi(R)$ is isomorphic to $R$, where $\varphi:R \longrightarrow J(R)$ is the Abel map. Hence, in this case the theta divisor $\Theta$ is also a compact Riemann surface of genus $2$. Hence the Jacobian variety $J(R)$ contains a compact Riemann surface $C:=\Theta- [k]$ of genus 2, and $(J(R), [C])$ gives a principal polarization. Let $E$ be a elliptic curve. Suppose that the self product $E \times E$ contains a compact Riemann surface of genus 2 and that $E \times E$ is isomorphic to a two-dimensional abelian variety $A$. In this case $(J(C), [C])\cong (A, [C])\cong (E\times E, [C])$; isomorphisms as principally abelian varieties. On the other hand, for points $a, b \in E$ a divisor $D = a \times E+E \times b$ is ample and $(E \times E, [D])$ is also a principally polarized abelian variety. But $(E\times E, [C])$ and $(E\times E, [D])$ are not isomorphic as principally polarized abelian varieties.

A first question is: For $(E \times E, [D])$ to be a principally polarized abelian variety, it should not be that $h^0(D)=1$? If so, how do I conclude that $h^0(D)=1$?

And another question is: Why $(E\times E, [C])$ and $(E\times E, [D])$ are not isomorphic as principally polarized abelian varieties?