Question

Does discharging always happen at cathode?

Answer 1

No.
In charged battery, the aggregate effect of the chemical reactions taking place between the anode and the cathode discharges electricity.
The anode undergoes an oxidation reaction and cathode undergoes a reduction reaction.
In simply words, the chemical reaction at the anode releases electrons and the reaction at the cathode absorbs them. When the electrical path provided by the electrolyte and an external electrical circuit connects the anode and cathode, the two simultaneous reactions proceed and the electrons freed at the anode travel through the external electrical connection and react chemically at the cathode to make the cell function.

I hope its clear that in discharging, both anode and cathode are involved, thus the statement is wrong