This is not the Fermi energy. It is just the dispersion relation for free electrons.
It may give the Fermi energy if you take k=k_F where k_F is the so called Fermi radius (but it's a radius in the k space, not in real space).
The difference between the 3D case and 2D case is in the way you calculate the Fermi radius.
The Fermi radius is the radius of the "sphere"that contains all the occupied electron states.
In 2D the "sphere" is a circle.
The dispersion relation is the relationship between wave-vector and frequency (k and omega) - for a wave. If the relation is linear then there is no dispersion (both phase and group velocities are independent of frequency).
By extension to a particle, the relationship between energy and momentum is sometimes called dispersion relation (for quantum particle, p=h*k and E=h*omega, so you can take it back to k-omega)
The dispersion relation is a characteristic of the medium in which the wave propagates (either EM wave or electron wave-function, or even elastic wave).