The issue is not the physical "joystick"; the term is being used metaphorically, to talk about who's in charge of what at any given point in time.
Picture a dozen people trying to play (simultaneously) one instance of The Legend of Zelda on an NES Classic with one controller. Every one of them has a different idea about where to go next, who to talk to, how to respond to the prompts, etc. But there's only one Link.
Ahh.. Thank God. That's much more reasonable of a reason for a dispute.