The sensitivity of the sensor does not change with camera settings. It is what it is per the design.
The designers and microelectronics people have been trying to keep semiconductor noise to zero. There are many factors, temperature is a big culprit. Note that noise signals are not constant, it's random in both the strength and quantity.
What changes with ISO is the threshold of the sensor output processing circuits. For low ISO, it sets the threshold at higher values, means light signals below a certain level are ignored and hopefully noise signals which are usually low gets cutoff as well.
Once you get into high ISO settings, the threshold get lowered and stronger noise signal sneaks through and shows up. You'll never escape noise. A good sensor is one that generates minimal noise and give high signal outputs for light that falls on it to that noise can be controlled.
A more sensitive sensor does not have better 'noise control' Its the stronger the signal strength that allows you to set higher thresholds (lower ISO) to capture the photo while ignoring the noise signals.
An analogy is oil floating on water. The oil is your signal, water is noise. Noise is a thin later of water, If the layer of oil is thick, you can cutoff the layer of water with some oil and still have a good layer of oil(picture) Sacrificing more oil also allows stronger noise signals to be cutoff giving a cleaner picture. If the oil is thin (signal), losing some of the thin layer leaves little, you'll have to work at the separation layer and noise.
That is also why fast lenses are better, with the larger lens, more light falls on the sensor and generates a stronger signal, allowing noise to be cutoff with minimal effect on the picture signal.