Clarify on recursively enumerable sets

Question

In a course of mathematical logic we said that if a (first order) theory $T$ has the standard model $\mathbb N$, every recursively enumerable relation on $\mathbb N$ is weakly representable in $T$. The proof is immediate, and it doesn't explicitely involve that $T$ has the standard model. I don't understand if that condition is only necessary because recursive functions and relations are defined on $\mathbb N$, or if there is something deeper. Thanks for your patience