Earlier quoted context omitted.
That's not a base unit, or a unit of any kind. What's it doing in a list of SI base units?
It is a base unit, a unit for lots of stuff, and it is not dimensionless it is 6*10^23/mol
There is a concept of "the Avogadro constant", which is defined to have units of mol^{-1} (at least, according to a cited statement on wikipedia), but that is not a coherent concept -- since mol is dimensionless, mol^{-1} is also dimensionless.
Just look at https://en.wikipedia.org/wiki/Mole_(unit)#Criticism :
> Since its adoption into the International System of Units in 1971, numerous criticisms of the concept of the mole as a unit like the metre or the second have arisen:
> the number of molecules, etc. in a given amount of material is a fixed dimensionless quantity
> the mole is not a true metric (i.e. measuring) unit
Or look at https://en.wikipedia.org/wiki/Atomic_mass_unit :
> One unified atomic mass unit is approximately the mass of one nucleon (either a single proton or neutron) and is numerically equivalent to 1 g/mol.
amu and g are both units of mass, so 1 amu = 1 g/mol is an explicit statement that mol is dimensionless.
Calling mol a unit won't accomplish anything except corrupting your dimensional analysis. mol is not analogous to the SI units meter, second, ampere, gram, kelvin, etc. -- it is analogous to the SI prefixes kilo-, mega-, milli-, micro-, nano-, etc.