Why is the field quantization applied to the free particle? This post claims that there is no real photon (particle) with a plane wave solution well-defined momentum state).
It makes sense somehow to me. I can think of several arguments:

*

*The plane wave solution doesn't have well-defined probability as it exists in infinite space with equal probability. We will never be able to detect those particles.


*Our probability interpretation breaks down with the plane wave solution. It's not square-integrable.
Even though it seems non-sense, we apply quantization rule to the free particle which makes quantization as the imaginary process which doesn't exist in reality.
Besides 'it works' argument, do we have any explanation why this quantization process makes sense?
 A: All models are wrong, but some are useful
Much of physics is inductive theories, which build description of complex phenomena from small elements. Neither photon, nor a point-like particle in free space (the basis of the Newtonian mechanics), nor a harmonic oscillator, nor plane waves exist. However, they are simple and convenient building blocks for developing theories that successfully model natural phenomena. Note that none of these theories tries to take account of everything - it would be meaningless - the question is that of keeping what is important, and omitting whatever is inessential.
A: 
Besides 'it works' argument, do we have any explanation why this quantization process makes sense?

Quantum field theory (QFT) is used for interacting elementary particles, and also as a tool/model for other quantum mechanical states (condensed matter for example).
Your premise:

we apply quantization rule to the free particle

Is wrong:
QFT was not developed to study free particles, because as you say the plane wave solutions cannot describe the probability of seeing a particle at ( x,y,x,t). To describe with fields a single particle one has to use the wave packet solution,.


Besides 'it works' argument, do we have any explanation why this quantization process makes sense?

The "it works" is reserved for the Feynman diagram use of the QFT tool/model, .
