You can start by looking at the paper by P.J. Hilton and W. Ledermann, "On the Jordan-Hölder theorem in homological monoids", where three axioms are needed to establish the decomposition, and the third one is essentially guaranteeing the second isomorphism theorem.

In "Mal'cev, protomodular, homological and semi-abelian categories" by F. Borceux, D. Bourn, there is a chapter devoted to homological categories (which are pointed, regular and protomodular), these are the categories where certain lemmas of homological algebra hold true (five lemma, nine lemma, snake lemma, Noether isomorphism theorems etc.). The fact that Jordan-Holder holds for these categories is proven in "Jordan-Holder, Modularity and Distributivity in Non-Commutative Algebra" by F. Borceux, M. Grandis.