I'm not sure how to reproduce it exactly, but I believe it has something to do with riding your mount. When I rode, the icon disappeared but my companion still followed. My companion still attacks and generates threat as normal, but enemies cannot attack him and simply stand next to him doing nothing. I cannot heal him nor see his health bar at all.
Comments
What I had done was rode my mount, entered into combat, and then dismounted. I believe those are the steps for making this bug happen, though I am not sure if other things have an effect too.