Person is quite common in conlangs, but we can even almost find other person-like structures in human languages - some Asian pronoun systems rather may be ranked by social stature than by speaker > listener > other.
We could imagine some other system, though - the simplest being some way of adjusting how social stature is determined (age, with gender as tie-breaker? gender, with age as tie-breaker? some other type of social structure than any known human structure); maybe to some not entirely 'ordered' type of ranking (so it's more of a social network than a social hierarchy).
Suggestion: come up with a different system than person, and post in a comment.