Suppose that $C_1, C_2$ are two curves of genus $g \geq 2$ defined over a number field $K$. Let $J_1, J_2$ respectively be their Jacobians. Suppose that $J_1, J_2$ are isogenous over $K$ and $C_1(K), C_2(K)$ are both non-empty, can $C_1(K), C_2(K)$ have different cardinalities?

For $g = 1$ and without the assumption that $C_i(K) \ne \emptyset$, the conclusion is obviously false. Take any elliptic curve $E$ over $\mathbb{Q}$ of positive rank such that the $2$-Selmer group of $E$ is non-trivial, so that there is a genus one curve $C$ which represents a non-trivial $2$-Selmer element of $E$. Then $E$ is isomorphic to the Jacobian of $C$, and the Jacobian of $E$ is itself, so that $C,E$ have isogenous Jacobians but $E(\mathbb{Q})$ is by assumption infinite but $C(\mathbb{Q}) = \emptyset$.