Wikipedia would have you believe that "parts per million" is a dimensionless coefficient, by a similar argument. It's not true; for example, grams of gold is not the same unit as grams of seawater.
Indeed, their argument is belied by the old intro to Chemistry technique I learned -- start with what you know, and keep multiplying by 1, or things that are true expressed in fractions, like 3.28m/ft, stuff like that. That example fact is context-less (although it has dimension), but most aren't, so it's not "gram" as a unit, but "gram of gold in solution", otherwise you will get completely screwed up (particular if you don't specify "in reagent X" as part of the unit context -- wowzers the errors can jump out at you pretty quickly).