What stimulus features determine the psychophysical power law exponent?
Mainly: the choice of variables.

Beware that the exponent depends on the choice of parameters; or even worse - relations can change from power-law to linear, logarithmic or exponential, with redefining variables.

E.g. you can measure sound volume in either amplitude, energy density (square of the amplitude) or dB (logarithm of the first one). None of it is privileged. Moreover, "psychological magnitude" needs exact definition, for the same reason (so not only to specify which type of reaction is that but what is the exact physical quantity being measured).

In general, if you consider only qualitative parameters, then you can look at only at qualitative behaviour of the relation (e.g. if it is monotonous). To calculate exponents (or even - to claim that a certain relation is a power function) it is not enough.

Of course, there are many adaptations to make sensitivity higher (e.g. edge detection) or lower (e.g. to see well in lighting conditions differing by order of magnitude, in terms of light intensity). But to compare their exponents, for the comparison to make sense, you need to have a very good justification for the right choice of variables.