Sorry but im not really sure to had understand what you mean.
Assuming a character mounting something, we know he have separete initiative with this mount except the mount is trained and he decide to control it. If he control it, the mount share its initiative with him, but it can only take the disangage, dash or dodge action.
The animal companion of a beastmaster share its initiative with the ranger because of its own rules, and this make a huge difference with other mounts in my opionion, as anyone CAN choose to control a mount for benefit of a shared initiative (as they normally dont have it), but as the beastmaster companion already share its initiative with the ranger, he dont need to control it for have shared initiative, because they already share it as a companion rule, even not controlling it they will keep sharing initiative.
As i said im not sure this is the right interpretation, but i found nothing against it and it seem have more sense to my eyes than be forced to control it even if they already got the one benefit of controlled mounts by other rule, or to say that if he choose to dont control it than it lose the shared initiative (that is right for anyone else, because they have no other source of shared initiative, but the beast master share its initiative with the companion indeed because thats his companion, not because he is controlling it).