Fisher Information is a way to quantify only the precision of the instrument. However, with sensitivity the quantity is intertwined. Let me explain.
Precision is the measure of repeatability of the measurement result. If an instrument is very precise then the value you will get as reading will be the same always and the likelihood function will be a sharp function at that value and the variance of it will tend to zero. Precision reduces if there is noise. It can be viewed as increase in variance of the likelihood function of the unknown measurement value.
The image below will be helpful to understand it.
another image which shows how precision is related to the variance of the likelihood function.
Sensitivity on the other hand quantifies how well the sensor responds to the given input. Suppose you have a electric pressure sensor which takes pressure as input and appropriately responds as increase in voltage as output. For example if the pressure sensor can take a max of 100 PSI and it gives 5 V as output. Then the sensitivity is 0.05 V/PSI.
Now, the Fisher information measure the curvature of the maximum of the log of likelihood function, which as we saw is measure of precision.
Now, suppose there are two sensors with same precision but different sensitivities and no noise filter. The one with higher sensitivity will also have noise amplification and reduce the precision of the instrument with higher sensitivity which can be seen as reduction in Fisher Information.
This is how Sensitivity is connected to Fisher Information.