It's a hierarchy matter.
Just imagine it were you. When you rotate your ankle, .... you'd be moving the bones from the ankle down. Your foot only, right?
When you wobble on one leg, it's not the ankle that is wobbling, but you wobble on your feet. Your entire body is moving right, left and forward while your foot is positioned stationary. Thus the body moves around the ankle. So you move your root node in your joint skeleton instead (usually the pelvis).

The IK-handle for the leg should be set to sticky if you don't have it keyframed yet (keyframing an IK-handle will automatically make it sticky). This should keep the ankle positioned while the body moves around.
However,.. when standing on one leg, the other leg should dangle with the body. But of course you have IK-handles there too. They will keep the ankle stationary like the foot on the ground but only in the air, what you don't want. You have to blend IK/FK for this dangling leg (see previous post).