There’s a study called generalized primes which investigates in a more general way the relationship between arbitrary elements called primes and the composites which can be formed from them according to specified rules of composition. Comparisons can be made among a variety of numerical systems or any orders of combinatorial species one might imagine. I seem to recall at least one old monograph by Rademacher on the subject.
My fascination with questions like that led me many years ago to the Riff and Rote trick, a special case of the Make A Picture trick. There’s a bit on that in the following article.