The theta function (the left-hand side of the Jacobi identity) uniquely determines the values of ${||\gamma^*||:\gamma^*\in\Gamma^*}$ counted with multiplicities, just by looking inductively at its asymptotic expansion as $t\to +\infty$: the leading term will be $1$ since the multiplicity of $0$ is 1, the next term will be $Ne^{-4\pi^2\alpha t}$, where $\alpha$ is the smallest norm and $N$ its multiplicity, and so on.
Similarly, the RHS uniquely determines the values of ${||\gamma||:\gamma\in\Gamma}$ by looking at the asymptotic expansion as $t\to 0+$.
So the two tori are isospectral iff they have the same theta function iff they have the same length spectrum.