For simplicity we assume that $R$ is an Artinian ring, such that any finitely generated $R$-modul has a composition series. To such an $R$-modul $M$ we associate the set of composition factors $C(M)$. We consider $C(M)$ as a multiset with respect to the multiplicities of the composition factors.
Then let $M_1, M_2 \subset M$ be submodules. Do we have $C(M_1 \cap M_2)=C(M_1) \cap C(M_2)$ (considered as intersection of multisets)?
$\newcommand{\ZZ}{\mathbb{Z}}$The inclusion $C(M_1 \cap M_2) \subseteq C(M_1) \cap C(M_2)$ may well be proper.
Suppose $R$ has two non-isomorphic irreducible modules $S, T$, and let $S_{1}, S_{2}$ be two copies of $S$. Consider $$ M = S_{1} \oplus T \oplus S_{2}, \quad M_{1} = S_{1} \oplus T, \quad M_{2} = T \oplus S_{2}. $$
As $R$ you may take the complex group algebra of a non-trivial finite group, for instance.