Earlier quoted context omitted.
I think the reason that using mouse is more precise is not because of its shape, but because we use at least all of our palm to use it while with controllers you only have one thumb to connect the stick with your body.
I disagree. I think the reason is more fundamental: with a mouse, you're doing a direct translation (distance in pad → distance on screen), while with a gamepad you're doing an indirect translation (distance in stick → velocity in screen). This is harder, because you're essentially forced to calculate (in a low conscious way, of course) distance/speed to know for how much time you should turn the stick.
There's no fundamental reason this has to be the case, of course; if the developer wanted to, they could use stick input as position rather than velocity. Conceptually, this is no different than using a high resolution trackball.