I don't know of any references that discuss other ionization types. 

The basic physical idea, though, is that you have to accelerate the electron, then radiatively recapture it. The HH photon energy comes from how much you can accelerate it while it's free, but you need the radiative recapture step to actually get that energy out as a photon. In most ionization processes you don't have any way to efficiently and coherently return the electron to the ion core. 

I'm new to the field and I'm sorry if I ask novice questions: I'd like to understand what is exactly the unique property of the tunneling ionization that lead to the generation of high harmonics within the three-step model. Why, for example,  the first step in the three-step model cannot be ionization of the electron by multiple photons or some other type of ionization? Is it because only in the tunneling ionization the ionization current consists of   sharp peaks with time interval equals to half the laser period? or is it because only in the tunneling ionization the electron wave function, used to calculate the expectation value of the dipole operator, is a superposition of the ground state and continuum state wave functions?-  I'd appreciate if you can explain this point further or refer me to a scientific source which explains these basic questions.  

You may try this:

http://psi.ece.jhu.edu/~kaplan/PUBL/AEK.pubs/84.pdf