Q:

What functions qualify to be called measures of dispersion by statistician?

What functions qualify to be called measures of dispersion by statisticians?
Why there are so many of these?

A:

You can find a nice explanation in the link I posted in your previous question:

A measure of statistical dispersion is a nonnegative real number that is zero if all the data are the same and increases as the data become more diverse. 

The reason why there are so many is the same I stated before: different measures provide informations about different features of a distribution. Some of them are based on quantiles, some others are based on moments ...

