Why a anode is negative and cathode is positive in galvanic cell

A galvanic cell represents a spontaneous reaction. You are correct in that oxidation always occurs at the anode, regardless of the type of cell. Since oxidation is the loss of electrons, an element (typically a metal) will be losing electrons and forming ions in solution at the anode.

As an example we have a cell made from two compartments in which there is copper metal incopper sulfateand zinc metal inzinc sulfate. The zinc spontaneously loses electrons to be oxidized to Zn2+ while Cu2+ are reduced to copper metal, which is the cathode. Zinc is the anode and copper the cathode.