Tribalism, presumably. Chik-Fil-A has intentionally made their brand about serving a specific tribe, being part of that specific tribe. That tribe has ideals and to work for that tribe is to live up to those ideals and if you don't live up to those ideals you are fired and probably not a "real" member of that tribe. It's tautological, but so is a lot of tribalism.

Given how much of that tribalism is also explicitly religiously coded, I find it's hard not to want to apply harsher words like "cult-like" to Chik-Fil-A, specifically, but "sect-like" is probably more accurate given how predominant both their business culture tribe and religious tribe are in American politics today even if "sect-like" doesn't have quite the same harsh connotations designed to help you question the systems of power in place.