Cameras are engineered in accordance with an understanding of what exposure is and how it's determined; an understanding that has foundations in science going back to the 19th century. The following is taken from the "Manual of Photography," a technical examination of cameras and the photographic process first published in 1890 with multiple later editions having followed. This excerpt is from the 6th edition (1971):
"The exposure received by a film is governed by the strength of the light falling on it and by the time for which that light is allowed to fall. We have already seen that light falling on a surface is defined as illumination. The relation between exposure (E), illumination (I) and exposure time (t) is expressed by the equation E = I×t."
You'll note there's no mention of ISO, ASA or any other measure of the sensitivity of the medium to light in the description of exposure or among the factors directly determining exposure.
To link this understanding with contemporary digital photography, we can reference ISO 12232:2019, which defines ISO's role communicating an exposure index to the camera:
"An exposure index (EI) is a numerical value that is inversely proportional to the exposure provided to an image sensor to obtain an image. Images obtained from a DSC [digital still camera] using a range of EI values will normally provide a range of image quality levels. The photographic sensitivity of a DSC is a particular EI value calculated from the exposure provided at the focal plane of the DSC that produces a specified camera image signal level. The EI value for a specific image captured by a DSC shall be equal to the EI reference exposure of 10 lx⋅s divided by the focal plane exposure used to capture the image,..."
Again, we see an understanding of what exposure is and how it's determined that is consistent with the definition in use since the 19th century. Modern cameras are designed, engineered and built to function in accordance with this long and well-established understanding.
The answer to your question, "An exposure of what?" can, I think, be summarized as electromagnetic energy. In most cases, it's energy from the visible light portion of the spectrum. However, I suspect an imaging device engineered to be sensitive to areas of the spectrum in the IR, UV, or radio portions of the spectrum would define exposure similarly while using the relevant energy unit.
If you seek a more technical explanation or understanding, I would encourage you to search posts by bobn2 and Iliah Borg. To narrow the list of results, you might try associating the name with ISO12232:2019 or some other topical terms.