smn-theorem: Application by instantiating s with other function
I think you are reading it backwards.

Usually you want to find $h$ such that $\varphi_{h(n)}(i) = f(n,i)$, where $f$ is a computable partial function. Further, you want $h$ to be total recursive (or even more: primitive recursive).

In order to prove that such an $h$ exists, as well as to concretely define it, one picks any index $m$ for $f$, so that $\varphi_m = f$, and chooses $h(n) = s(m,n)$.

Note that, when reading the reasoning above in the other direction, it becomes wrong. That is: it is not the case that, since $\varphi_{h(n)}(i) = f(n,i)$ we can conclude $h(n) = s(m,n)$.

Indeed, $h$ may also be another function, like e.g. $h(n) = pad(s(m,n))$ where $pad$ is a padding function (which returns the Gödel number for an equivalent TM, such that said Gödel number is larger). Infinitely other (computable) $h$ exist.