Do there exist two distinct entire functions $f(z)$ and $g(z)$, neither of which is identically constant and such that f / g is non-constant*, such that $$f(m + n\:i) = g(m + n\:i)$$ for all $(m, n) \in \mathbb{Z}^2$?
Certainly, if two such functions exist, then at least one has an essential singularity at infinity. Indeed, define $h(z) := f(1/z) - g(1/z)$, which is holomorphic on $\mathbb{C} - \{0\}$. Since $h$ is zero on a sequence converging to the origin, it is either holomorphic there (implying $f\equiv g$ by the identity theorem) or has an essential singularity: a pole is not an option since that would imply $|f(w_n)|\rightarrow \infty$ for all $w_n$ converging to zero. Therefore, either $f$ or $g$ has an essential singularity at the origin.
I have a feeling that either Picard's Great Theorem or Weierstrass factorization, both of which I have only passing knowledge of, could be useful. Any ideas?
(PS: I did a search before posting and see plenty of answers on the existence of such a function, but none so far on its uniqueness.)
*The bolded condition has been added because, as user Conrad points out, we can simply scale $f$ up by a multiplicative constant.