Why is the oxalate ion a bidentate ligand? Why does it not act like a tetradentate ligand (having four oxygen atoms, with lone pairs)?
The oxalate anion cannot be tetradentate. While there are four oxygen atoms, the ion is planar and the oxygen atoms are pointing in different directions.
The best any metal can do is bond to two of them, although there are two bonding modes possible:
Perhaps you were envisioning some $\kappa^4$-bonding mode that is "side-on". However, most of the nonbonding electrons (orbitals) are also in the plane.
This is not to say that oxalate cannot form four bonds to metal centers; it just cannot do so with only one metal center. Oxalate can act as a bridging ligand