Why is the mole a unit of measurement?
I think you are confusing unit of measurement with fundamental unit.  A unit of measurement is a definite magnitude of a physical quantity.  Consider a very common grouping unit, the dozen which is conceptually the same as a mole and is a little easier for us to grasp since it is a commonly used small number.  As far as I know, the only requirements for a unit of measurement to be valid is general consensus.  If you and I agree that we will measure distance in units of 'Chemistry Textbook Thickness', then it is a valid unit of measurement.

A fundamendal unit is a part of a particular set of units upon which all other units can be derived.  In this case, the mole is related to the fundamental unit of mass and is defined as an amount of items equal to the amount of carbon atoms in 12 grams of $^{12}C$.