Looks like he does Mark, it looks red, unless the shaft is red?
You could always shim it, but that kind of movement is usually a bearing inside the gearbox, or maybe the little plug thing has come out of the other side of the gearbox, or it's got worn.
It might be just my eyes, but towards the end of the video, you can see the bearing moving back and forward inside it's recess in the gearbox, so I'd say it's just the plastic wearing on your gearbox.
Personally as long as there's no up/down movement, I wouldn't worry.
The gears will still mesh and if it still spins freely and it won't foul anything either, I'd just run it.
Failing that, whip the gearbox out and check for wear in all the bearing recesses.