For a very general, principally polarized Abelian variety $(A,\Theta)$ of dimension $g$ over $\mathbb{C}$, every Cartier divisor $D$ on $A$ is numerically equivalent to $m\Theta$ for some integer $m$. In particular, the intersection number $D^g$ is $m^g \Theta^g$. So the minimal degree is $g!$, not $2$.

For a very general, principally polarized Abelian variety $(A,\Theta)$ of dimension $g$ over $\mathbb{C}$, every Cartier divisor $D$ on $A$ is numerically equivalent to $m\Theta$ for some integer $m$. In particular, the intersection number $D^g$ is $m^g \Theta^g$. So the minimal degree of an effective, nonzero divisor is $g!$, not $2$.