What is active mass?
The term "active mass" is a historical term.

The concept of an equilibrium constant was developed by Cato Maximilian Guldberg and Peter Waage.  The Law of Mass Action has also been referred to as the Law of Guldberg and Waage, historically. 

Guldberg and Waage defined the term "active mass" in the 1867 Études sur les affinitès chimiques.  


  l'on peut choisir un volume arbitraire, par exemple un centimètre cube. Les 
   quantités d'un corps qui se trouvent dans lcc du volume total se nomment la 
  masse active du corps. 


or roughly translated:


  One can choose an arbitrary volume, for example a cubic centimeter. The
  amounts of a body located in 1cc of total volume are called the active mass of the body. 


And as stated in The Laws of Reaction Rates and
of Equilibrium J. Chem. Ed. 1956, 33 (4), p 178 :


  Guldberg and Waage in their study of chemical
  affinity formulated the law of the rate of chemical reactions
  as proportional to the "active mass" of the reagents.
  This they called the law of mass action and
  very clearly defined "active mass" as amount per unit
  volume. (emphasis added)


So, from a historical point of view, it is false that "active mass" is dimensionless.  

However, in relating Gibbs energy to the equilibrium constant, it is necessary for the equilibrium constant to be dimensionless, because only the logarithm of a dimensionless quantity is logical.  For this reason, the text mentioned in the OP redefines "active mass" to be dimensionless.