It is for the upper lip, but not for leading around.
It is called a "twitch" in the US and UK, and it is used as a method of restraint for a very long time.
You clamp it down over the horse's upper muzzle (not too tight). It used to be thought that it was just a distraction but later research indicated that there is an interesting calming response that is induced from the stimulation of the nerves in this area.
Twitches used to be simply a short loop of rope or chain attached to the end of a short pole, but those were mostly replaced by the clamp-type.