A big part of clan authority seems to depend on sheer, terrifying power, honestly. The kind of power that can literally squash a challenger without a second thought. That establishes a baseline of fear, but the lore I've read suggests it's not enough for long-term control. Loyalty needs something to hook into.
Ancient blood magic pacts often come up. The idea that the king's blood is the original source or holds a magical bind over those he's turned creates a biological and mystical hierarchy. In 'The Vampire Lestat', the Maker holds immense sway, not just from age, but from that primal connection. It's less about charisma and more about a supernatural leash.
Then there's the old-school feudal model: providing protection, territory, and purpose. The king maintains order, settles disputes, and keeps the clan safe from hunters or rival factions. In return, the clan offers fealty. It breaks down if he fails to deliver that security. A king hoarding all the good feeding grounds or leaving his people exposed won't stay king for long, no matter how ancient he is. The moment a stronger, more competent leader shows up, the clan's allegiance can shift frighteningly fast.
The most interesting dynamic I've seen is when control is psychological. A king who cultivates devotion, who makes his followers believe in his vision for their kind, or who positions himself as a father figure or a dark messiah. That's the kind of control that survives a temporary loss of power. It's messier, but way more fun to read about than simple brute force.