I feel there should be an option for the player to allow their companions, particularly those of noble backgrounds, to form their own clans but it should cost like 200 influence and you should have 10-20 relations with that companion.
Unless from a distinguished noble family, they will generally start off as a level 1 clan in your kingdom with one member but eventually level up the clan, recruit their own companions, marry and have kids, ect, especially if given many fiefs for income.
Not sure if I want that same mechanic for AI clans though, might need to design another one. The AI Kings could be able to do something similar but currently no AI clans have any companions to grow their clans at all. They are all lords with parties on the field or governors, no caravan leaders, or even companions to heal or fight in the clan leader's party.