Is it possible to construct a multiplicative group from $\mathbb{Z}_n$ if $n$ is not a prime number? With $n$ being a prime number I know we can generate groups over multiplication. Is it possible the other way around ($n$ not being a prime)?
Yes. The subset of $\Bbb Z_n$ of integers that are relatively prime to $n$ (otherwise said: the set of $x$ in $\Bbb Z_n$ with $\gcd(n,x)=1$ ) form a group under the multiplicative law of $\Bbb Z_n$ (multiplication modulo $n$ ).

That finite group is noted $(\Bbb Z_n^*,*)$ or just $\Bbb Z_n^*$, or $(\Bbb Z/n\Bbb Z)^\times$. It has $\varphi(n)$ elements, where $\varphi$ is the Euler totient.

That group is cyclic (otherwise said: generated by a single element) iff $n$ is in one of three forms: $4$, $p^k$, $2p^k$ (for $p$ an odd prime and $k\in\Bbb N$, including $k=0$); that is for $n$ in A033948.

Thanks to Poncho for a useful comment.