The horse and follower’s custom distance resets back to default on each login (single player). Not sure if it does the same on server reset (multiplayer), but is likely. You can test single player by starting game, admin horse, set follow distance to longest distance, confirm change was successful and then log out of game. Once logged back in the horse is following at 5 meters again.
This begs the question as to whether or not the other settings (attack and chase distance) also reset. Although I suspect they do, I have no way to test that.