If the picard groups $Pic_K^0(C_1)$ and $Pic_K^0(C_2)$ associated to the curves $C_1, C_2$ are isomorphic (as abelian groups), then the curves are not necessarily isomorphic
I don't know if there are only counterexamples or a generalization why it does not hold, but the converse should be valid, i.e. if the curves are isomorphic, so their picard groups. (this is an exercise in elliptic curves)
In the book of Lawrence Washington (Elliptic Curves: Number Theory and Cryptography) the map, $Div^0(E)/(\text{principal divisors})\to E(\overline K)$ is called to be an isomorphism, in other sources only a bijection, which one is true ?