Is the "modular subset product" problem NP-complete?
The claim that you linked to (that subset product is strongly $NP$-complete), is incorrect: it is only weakly so. The numbers $s\in S$ are small and can be represented in unary, but $t$ has to be represented in binary for the reduction to work.

If you consider it over $\mathbb{Z}_n$ it remains weakly $NP$-complete, since we can just take $n$ to be $\Pi_{s\in S} s$. In this case, the fact that we're doing modular arithmetic becomes irrelevant. $n$ will be exponential in $\Sigma_{s\in S} s$ so that $\log n$ (the size of a binary representation of $n$) will be polynomial in the size of the original instance (in which all $s\in S$ were represented in unary).