In your example, it's sooner a matter of collocations (set phrases/word combinations) with the word "balance" as in "the balance of animals and plants in the environment, the balance of advantage/forces/power".
There's no usage of the preposition "among" after the noun "balance", if that is what you mean. So, "the balance between a, b and c" is correct.
To see the difference between "between" and "among", you've already gotten the link, which is in the comment to your question.