Given two *complex tori* $X_1$ and $X_2$, there is always a canonical *isomorphism*

$\widehat{X_1 \times X_2} \cong \widehat{X}_1 \times \widehat{X}_2$,

see for instance Birkenhake-Lange's book *Complex Abelian Varieties*, Exercise 11 page 43.

Indeed let us write $X_i=\mathbb{C}^{g_i}/\Gamma_i$, where $\Gamma_i$ is a lattice. Then

$X_1 \times X_2 \cong \mathbb{C}^{g_1+g_2}/ \Gamma_1 \times \Gamma_2$

and, by standard representation theory of abelian groups, the character group of $\Gamma_1 \times \Gamma_2$ coincides with the direct product of the  character groups  of $\Gamma_1$ and $\Gamma_2$.