I am aware that energy mass are inter-convertible using the famous $$E=mc^2$$
But why is it that energy and mass are basically the same thing that takes different forms?
I am looking for a theoretical answer rather than answer based on formula. I am also changing matter to mass as pointed out in the comments.
The $m$ in the famous $E=mc^2$ is “mass”, not “matter”. Mass and energy are properties of matter, but they are not the only properties. Matter also has other properties like spin and charge.
Regarding mass and energy. Although super-famous, the formula $ E = mc^2$ is a simplification of a more general formula: $E^2/c^2-p^2=m^2 c^2$. The famous formula only applies for the special case of $p=0$. In general mass and energy are different, but they are related to each other and to momentum by the more general formula.
Specifically, mass, energy, and momentum are all parts of the relativistic four-momentum. Energy is the component of four-momentum in the time direction and momentum is the component of four-momentum in the space direction. Then mass is the magnitude of the four-momentum.
This is because of SR (in the following $\eta_{\mu \nu}=diag(-c^2,1,1,1)$):
We have that for the four-velocity $u$ of a massive particle it holds:
$$u^\mu u_{\mu}=-c^2$$
Since the 4-momentum is $p=mu$ we have:
$$p^\mu p_\mu =(m u^\mu)( mu_\mu)=-m^2c^2$$
But since $p=(E/c^2, \vec{p})$:
$$p^\mu p_\mu =-c^2\frac{E^2}{c^4}+|\vec{p}|^2=-m^2c^2$$
I.e.:
$$E^2-|\vec{p}|^2c^2=m^2c^4$$
That reduces to $E=mc^2$ in the rest frame of the particle (where the three-momentum $\vec{p}$ is null).
One possible way to look at it is that we tend (anthropomorphism) to regard processes as being fundamentally different from objects and to class physical phenomena into these two categories (objects, processes), as if these two categories were inherently mutually exclusive. Quantum mechanics teaches us that the particle/wave duality is erroneous; electrons, photons have both characteristics of a particle and a wave.
Another example of physical phenomena which appear at first to be objects but prove to behave in a more dynamic fashion on closer look is given by the example of protons or neutrons. Most of the mass of the proton (or neutrons) does not result from the sum of the masses of the quarks but from the average kinetic energy of these particles. So basically, most of the mass of the p/n is kinetic energy; therefore, most of the mass of nuclei is kinetic energy, and by extension, most of the mass of atoms (or matter more generally) is kinetic energy. https://www.forbes.com/sites/startswithabang/2016/08/03/where-does-the-mass-of-a-proton-come-from/?sh=791273ea2e1d
Mass, therefore, is essentially energy; stated otherwise, mass is just an anthropomorphism, a macroscopic designation, just a word for a more complex, dynamic phenomenon. To put it another way, one couldn't convert mass into energy and conversely, if at some level these two facets of matter were not one and the same thing.
