Why is the mole a unit of measurement? I really can't understand why is the mole a unit of measurement and not just a constant. I mean, we use units to express things that we can measure but that are not countable, that are continuous - and not discrete. However, the number of atoms (or whatever) of something is clearly countable, discrete. It is also a huge number, which would request for a constant, a number to multiply to: Avogadro's constant is that number, like mega or kilo in SI for other units. We could still write 14 mole, where mole is the Avogadro constant (exactly the way it is), but every book I read so far states that the mole is a unit of measurement, official in the S.I., and so does Wikipedia. It just makes no sense to me that it is considered that way, but there is obviously a reason why... What is it?
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 $^{12}C$.