Characteristic and moment generating function of a random variable interpretation
As you well said, these functions are useful just for their mathematical properties that allow to compute the m-th order moment of a random variable. 

Additionally, they happen to match Fourier transform theorems. However, not in the sense of a time-domain signal, since the variable in the probability density function (pdf) is the random variable itself, and not time.
A frequency interpretation only makes sense in time-dependent functions. Actually the definition itself of frequency is how often something happens.

Anyway, one could consider the "bandwidth" for $\Phi_X$ in the $\xi$-domain as how spread the pdf is around the mean.
A more spread pdf would imply a narrower "bandwidth", and less spread of the pdf would imply a wider "bandwidth".
For example, consider a uniformly distributed random variable:
$$f_X(x) = 1/\delta~\Pi(x/\delta)$$
$$\Phi_X(\xi) = sinc(\delta~\xi)$$
As $\delta$ gets smaller, the random variable is constrained in a narrower interval, while the "bandwidth" in $\Phi_X(\xi)$ gets wider, and vice versa (measuring the "bandwidth" as the width of the main lobe, $2/\delta$, as we do for communication signals).

Just by observing the characteristic function one may have an idea of how spread is the pdf. By comparing two characteristic functions with different "bandwidths" one may determine which pdf is spread the most.