You actually support your sax at four points.
1. The strap that hangs your sax from your neck.
2. Your left thumb; rests against that (usually black) cap. You also use this thumb to operate your octave valve.
3. Your right thumb; is more or less in that bracket/hook. It looks like you have to lift your sax with that, but you push the sax a bit away from your body. (see more about this in BLOG 3)
4. Your upper teeth; they stand firmly on the mouthpiece. When you push with your right thumb, you will feel the mouthpiece being pressed against your teeth. (see BLOG 10 – mouthpiece stickers)
With those four support points, you keep your sax in place. Your eight fingers can now move freely without your sax immediately flying in all directions.
TIP:
Stand with your sax relaxed in play position, your mouth peace gently against your teeth, your thumbs in place, right thumb pushing the sax a bit away from you; and the other eight fingers loosely above your valves. So the sax is not really fixed now. Lower your under jaw and try to talk a bit. If you notice that you lose control, you can try if it helps when you hang your sax a little higher or lower; or push a bit more with your right thumb so that the mouthpiece is pressed a bit more firmly against your teeth.
Of course, it takes a while before you have good control over your sax in terms of balance. It will come.