I know that in a galvanic cell anode donates electrons through the external circuit. Is that the same electrons that the electrolyte gives it ?
Anode is the terminal where oxidation occurs.
In galvanic cell, anode is the negative terminal while in electrolytic cell anode is the positive terminal.
And the electrons you are talking about are the free electrons which are liberated in the oxidation half after the oxidation.
Consider: m A + n Bm+ -> n B + m An+
Actually what happens is in the oxidation half cell, electrons are liberated because of the oxidation reaction and in the reduction half reaction, for the reduction to take place of the B metal, the required electrons are pulled from the oxidation half container (which are liberated during oxidation of atoms A) via wire which joins the two electrodes.
So basically electrons which flows in the external circuit are the electrons liberated from the oxidation of atoms A in the oxidation half.