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?