Dear,

What do you mean by "characterize" ?

I mean uniquely identifying underlying distribution.

The answer is Yes, if you're characterising the fisher information using the parameters of the PDF (mean and variance or up to some moment), then yes. But if you tune in the parameters say:

Gaussian : mean_g and var_g 

Another law: mean_l and var_l 

The fisher information might return the same numerical value. 

But in terms of FIR in terms of moments, i'm quite sure it's unique.

But if you have location family then Fisher information will be a constant.