How does an electron in the same shell of an atom have same energy despite of having different magnitudes of orbital angular momentum? For example, consider the $n=2$ shell. This will have $l=0,1$ values, so correspondingly there are 2 angular momentum states having magnitude $0$ and $\sqrt{2}\frac{h}{2\pi}$. So how do these 2 different angular momentum states have same energy? And how do we interpret this angular momentum?
 A: Are you referring to hydrogen?  In other atoms  different $l$ values with the same principal quantum number $n$ will have different energies. It is only for hydrogen with its single electron that all  states with the same $n$ have the same energy. In this sepcial case  there is an extra, hidden, $O(4)$ symmetry possesed by the Kepler $V(r)=k/r$ potential. This symmetry is absent for other atoms because the charge of the other electrons partially screens  the potential as you move away from the nucleus.   Consequently  states with lower $l$, which  penetrate closer to the nucleus, have lower energy.   
The Wikipedia article on the "Aufbau Principle" is a good source for this.
A: There is no particular reason to expect that energy and angular momentum should have a one-to-one coupling.
If we take the case of classical two-body orbits as a rough guide, we see that each (bound) orbital energy is associated with a upper limit on angular momentum but that the acutal angular momentum can range from that limit all the way down to zero.
And the situation with quantum orbitals of hydrogen-like (i.e. two body) systems is similar. Each (bound) energy is associated with a set of allowed angular momentum values that starts at zero and run up to some maximum. Nicely parallel to the classical case.
A: I would like to add for completeness  that there is fine structure in the spectrum of hydrogen, and atoms in general. It is only the simple potential models that do not see a difference due to angular momentum.

When the familiar red spectral line of the hydrogen spectrum is examined at very high resolution, it is found to be a closely-spaced doublet. This splitting is called fine structure and was one of the first experimental evidences for electron spin.



The small splitting of the spectral line is attributed to an interaction between the electron spin S and the orbital angular momentum L. It is called the spin-orbit interaction.

There is also hyperfine structure:

In atoms, hyperfine structure arises from the energy of the nuclear magnetic dipole moment interacting with the magnetic field generated by the electrons and the energy of the nuclear electric quadrupole moment in the electric field gradient due to the distribution of charge within the atom. 

So it depends on the potentials used in the solution of the quantum mechanical equations for the given atom.
Experimentally, both fine and hyperfine structure are observed if one takes data with enough accuracy.


Schematic illustration of fine and hyperfine structure in a neutral hydrogen atom

