How to write 'number of ...' in a unit?
In my experience $[m^{-3}]$ or $[cm^{-3}]$ as number density is very well understood throughout the communities of both earth, solar-system as well as exoplanetary scientists.

Take for example the recently published The atmosphere of Pluto as observed by New Horizons which in fig. 3 shows the derived atmospheric composition of Pluto from occultation measurments in $[cm^{-3}]$ with some model curves:

 (Property of G. R. Gladstone et. al 2016 and the New Horizons Science team)

The density in this unit is just a very natural and convenient outcome of the used occultation-inversion techniques. Thus of course the units are sometimes simply the result of the context in which a work is done, and as other commentators stated, a mole could be used.
However I'd argue that $[cm^{-3}]$ is sufficiently simple to be understood in any context and adding moles just makes things unnecessary confusing as it suggests numbers to have a unit, which they don't.
It just physically makes no sense.