Isn't it possible for a matter to have mass of an electron but opposite charge? I read that "Positrons" have the same mass as of an electron but it is oppositely charged but we don't call it a normal matter we have given a special name to it "anti matter". But why ? Why can't a matter particle have those properties ? Why is it classified as an "anti particle" and not a normal particle with lesser existence ?
 A: All particles have an associated anti-particle. Note that some particles, a.o. photons, are identical to their antiparticle. Which one of these are called particle and which antiparticle is based on natural abundance. Positrons, antiprotons, antineutrons are hardly found in nature so are termed antimatter. An electron is ascribed a positive lepton number. Henceforth particles with a negative lepton number get the prefix 'anti-'. Similarly baryons have a baryon number derived from the number of quarks and antiquarks. +1 is matter , -1 is antimatter. I don't know the convention for mesons.
A: 
I read that "Positrons" have the same mass as of an electron but it is oppositely charged but we don't call it a normal matter we have given a special name to it "anti matter". But why ?

It is a long story, the story of building up the current model for particle physics interactions at the level of the microcosm of atoms/molecules/lattices. It is called the standard model and is successful in fitting the existing data, (with some exceptions) and predicting new situations.
The concept of antiparticle is inherent in the model, for the table of elementary particles

there exists the exact same for antiparticles, both assumed axiomatically. Antiparticles have the negative quantum numbers to the particles. Composite antiparticles, as the antiproton, exist symmetrically to the composite particles.
The basis is special relativity, where when particle meets antiparticle, pure energy is left, all the quantum numbers adding up to zero.
Thus when an electron meets a postitron two photons come out , because the lepton numbers add up to zero and the energy is available to create the photons. It is too small for larger mass particles.
When proton meets antiproton, again all the quantum numbers add up to zero, but this reaction has enough energy to produce a plethora of particles and particle antiparticle pairs, again  with the summed quantum numbers being zero.
At CERN a collider of proton antiproton was used to study the high energy interactions.

Why is it classified as an "anti particle" and not a normal particle with lesser existence ?

It is just terminology because the masses are equal but their quantum numbers are opposite . Antiparticles are treated the same as particles in all the calculations, while keeping track of quantum numbers.
A: Because it's not just mass that's identical, all properties of an electron are shared by the positron except opposite sign for charge (and also handedness).
