My understanding of ISO is that it is electronic gain from recorded light in a sensor, ie it doesn't actually change the sensitivity of the sensor (because that is dictated by the quantum efficiency of the sensor), rather it just amplifies the light more or less by reducing the dynamic range. As such, it doesn't actually influence exposure.
This being said, most sites/ books I have read on the topic refer to the 'exposure triangle' being f/ratio, shutter speed and ISO. If it is true that ISO doesn't actually change the exposure, why is it counted as part of this graphic? The only reason I could think of, is that exposure is defined in these cases, as the point at which the image is closest to middle grey, rather than the actual amount of light collected, and therefore signal from the sensor.