The Book [Auslander, Reiten - Representation theory of Artin algebras] begins with the Jordan-Hölder theorem for modules of finite length over arbitrary rings. The proof is probably quite standard - here is the idea:
Define the length of a module $M$ and the multiplicities of its composition factors as minimal length and minimal multiplicities over all (generalized) composition series. Then show that these functions are additive with respect to short exact sequences. The Jordan-Hölder theorem now follows easily by induction on the length of $M$:
For $l(M) \leq 1$ the statement holds clearly. If $l(M) \geq 2$ there is a submodule $0 \lneq U \lneq M$. Any (generalized) composition series of $M$ splits into a (generalized) composition series of $U$ and of $M/U$. By induction hypothesis, those sequences satisfy the claim, i.e. they have length $l(U)$ and $l(M/U)$, respectively, and certain factor multiplicities defined by $U$ and $M/U$. By additivity of the length function and the multiplicity functions shown before, the claim also holds for the chosen composition series of $M$.
I wonder whether this proof can be adopted verbatim to prove the Jordan-Hölder theorem for groups. At first sight, I see no reason why this cannot be done. However, I haven't seen this proof in any source concerning groups (usually, the Zassenhaus lemma is used instead).