Now, my question is why most camera apps use a limited set of shutter speed values instead of a continuous value.

Using discrete values rather than a continuous range makes it easier for photographers to select the right value, predict the result, and reproduce any given setting. The usual range of speeds provides enough granularity to achieve any exposure level the photographer is likely to want, so there's little benefit to being able to choose an arbitrary value along a continuous scale. The scale is (approximately) exponential, with the difference between any two full stops being a factor of 2, and that corresponds to the scales for aperture, sensitivity (ISO), and flash power, so it's easy to understand how to compensate for a change in one of the parameters. Read up on the exposure triangle if you're unfamiliar with that concept.

or due to some other factors, e.g., the support of underlying CMOS sensor?

The range of shutter speeds in use today is about the same as what you'll find on a camera that dates back 75 years or more.

I notice that the iOS API provides a function that takes a continuous exposure duration as input.

Tiny cameras like the ones in iPhones have fixed apertures, so the only parameters available for controlling exposure are shutter speed, ISO, and the LED "flash". Also, the exposure parameters are mostly controlled by the computer rather than by the photographer, so there's no reason to use a set of discrete values. That's a very different situation from what you have with a dedicated camera that's meant to be controlled by the photographer.