It's convenient for shutter speed, aperture, and ISO to operate in stops, because then it's easy to interchange them. Minus one stop of shutter means plus one stop of aperture or ISO, for example. Using fractional stops adds some versatility, but at the expense of having to cycle through more options. My DSLR lets me choose whether I want operate in third or half stops for that reason.
For auto modes, there's no really compelling reason to require certain speeds anymore. It's probably measured in milliseconds by the firmware anyway. For ISO, the steps are sometimes determined by hardware assumptions, but likewise there's no real reason the chips couldn't br made do different amplifications. However, for both of these, there's usually not a compelling reason to change how it's done. There's just not that much difference in less than a third of a stop of exposure. For aperture, though, some modern lenses in some camera systems are made for video and have smoothly variable aperture, so that exposure and depth of field transitions in a film aren't sudden jumps. (I'm not sure if these lenses can be used in such a manner in still photography mode, though. I wouldn't be surprised if the couldn't, again because the benefit would be small.) In any case, this is often not even possible with the current design of existing lens mounts which tend to communicate aperture in a primitive way.
It's worth noting that some camera companies reserve selection of fractional-stop ISO or shutter speed settings for their higher-end models. There is no technical reason for this. It's just that the full stops are perfectly adequate for almost all situations, and the increased flexibility is used as an incentive to push people into buying the more expensive models.