The fermi temperature is defined as $$ k_BT_f = E_f$$
But the fermi energy is the energy at $T=0$, where the energy level is the highest occupied for electrons. So, why is the fermi temperature defined as $\neq 0$?, What temperature $T_f$ is measured? Over who is T measured?
The temperature is set to zero to define Fermi energy because $T=0$ corresponds to the ground state of this electronic system. The electrons are still moving around with their zero-point motion, and the Fermi temperature corresponds to this motion. Considering that electrons don't contribute much to the heat capacity of a bulk material, the electrons can have relatively high kinetic energies without appreciably raising the temperature of the total system.
In a semi-classical description, if we think of the temperature as being related to the kinetic energy and therefore velocity of the electrons, the Pauli exclusion principle disallows the electrons to all be in the same state, meaning that the vast majority of electrons must have nonzero kinetic energy (since they can't all be in zero kinetic energy states, or more properly they cannot all have zero momentum). Therefore even at zero temperature (the lowest energy state of the entire system) the electrons are still moving, the energy of the system is not zero. The Fermi energy is the energy of the fastest electrons, and the Fermi temperature is the temperature that corresponds to to this energy by the same formula you provided.
