Why do electric fields and magnetic fields behave as waves or particles in certain property? The idea of electric field was initially developed as a substitution to force vector, we then attached energy associated with the system with them as we didn't had much of a choice to explain the "storage of energy in the system".
But, it's getting more weird now when we have momentum and stress in fields, these are general properties of particles and waves around us everyday.
Why is it so?
Are these just mathematical convictions or is there some real physical sense behind them which I am not able to grasp?
Or, are we stepping next to claim that charges produce some quantity which we cannot perceive directly in any manner but have wave/particle like characteristics?
 A: You do not indicate your level of physics education.
Waves were first seen in water, then the wave equations that described water waves were used for acoustics, and then when Maxwell brilliantly showed that light was electromagnetic waves, the expectation was that the electromagnetic waves  "worked" as far as energy and momentum goes, the same  way water waves and acoustic waves work: a medium carrying the energy and momentum. They invented the luminiferous  aether . 
This was the expectation until the Michelson Morley experiment that tried to measure electromagnetic waves on the aether, and found out there is no aether, but the waves were self propagating. The Maxwell equations held, but the energy was carried by the electric field and the magnetic field describing the wave.
That is where things were until the quantum mechanical revolution, where  they used wave equations to describe the probability of finding a particle at an (x,y,z,t), and the confusion of the particle wave duality comes in. In quantum mechanics, particles are point particles, or systems of point particles, described by four vectors.
It can mathematically be shown that the superposition of the wave functions  of very many photons, build up the classical electromagnetic field, but it needs quantum field theory to understand the derivation.
Mainstream physics now accepts that the underlying level of nature is quantum  mechanical. Thus the classical waves we see as light are composed out of zillions of photons and it is the photons that carry the energy and momentum. The quantum mechanical mathematics can also be used to describe static fields as described here .
This is the state of understanding of particles and interactions in mainstream physics, and the theories are predictive and are validated by innumerable data. There is no confusion that cannot be resolved by careful mathematics.
A: If I understand your question correctly, then it won't help to show you how one can write expressions for momentum and energy in terms of the electric and magnetic fields, and tell you that measurements of those quantities in these fields agree with the predictions obtained from such expressions. So, instead I give you the observation of a simple experiment that you can perform and then think about its implication.
A permanent magnet has a magnetic field around it. If I hold the magnet above a small metal object that is lying at rest on my table, then the object can be attracted toward the magnet and get stuck to it. Now think what happens there. For the object to be lifted upward against the attraction of gravity, it must gain some energy. How did that energy get to it? It must have been transferred via the magnetic field. Hence the field must be able to contain energy. Moreover, the object must have picked up some momentum while it is moving upward. The conservation of momentum principle tells me that the field must have transferred this momentum to the object. So the field must also have the ability to convey momentum.
