Why is the mol a fundamental physical quantity?

The mole definitely isn't a fundamental physical quantity. It's just a shorthand for Avogadro's number, to make really big numbers more tractable. It's purely there for convenience, there's nothing fundamentally physically significant about it.

Mols are a units of quantity. Technically, you can have a two cars, or a mole of cars, two forks, or a mole of forks, two baby rabbits, or a mole of baby rabbits. But since one mole is such a large number, it is only really useful for things that you have lot of, like molecules. In that case, though, it is very useful, since saying one mole is lot faster than enunciating 602 sextillion, or $6.022140857\times10^{23}$.

It is very important to know how many molecules of a particular type there is (for instance) in a beaker. If you have two highly reactive molecules in a beaker, it's probably not too dangerous: these two molecules will only destroy two of the floor's molecules were the beaker dropped. However, if you have a mole of these dangerous molecules, the floor might start to complain.

Mass and mole are completely different things: a mole of cars will weigh more than a mole of H2 molecules.

Is it a fundamental number in nature? It's (currently) a number resulting from atomic structure (fundamentally defined by the masses of quarks, Planck's constant and the way quantum mechanics works) and our definition of the gram, which is based on the international Kg prototype. Avogadro's constant is currently defined by experiment, and therefore has no absolute "right" number, just an agreed working definition.

This is a messy way to define things though, and there are many arguing that the Kg should be defined in terms of a particular element and Avogadro's number, which would put it on a more "fundamental" level in my book. (See https://en.wikipedia.org/wiki/Kilogram#Avogadro_project )

This would mean "fixing" Avogadro's constant by simply picking a number, then defining the Kg in terms of this, in the same way the second was "fixed" in terms of the so-many-oscillations of a particular frequency of light, rather than being a 60th of a 60th of a 24th of one rotation of Earth (a messy, variable number).