The companion stamina system is the yellow bar under your companions HP, and determines whether the companion follows you at the same speed or not.
It is ultimately a pointless system. I’m not sure of a way to improve it, because that would be investing time on an artificial problem that nobody likes in the first place.
So I would provide this feedback;
Remove the stamina system from companions, completely. It is pointless, and purely inconvenient. Companions need a lot of work, this is a good place to start.