Asteroid (16) Psyche is the largest M-type asteroid in the main belt and the target of the NASA Psyche mission. It is also the only asteroid of this size (D>200km) known to be metal rich. Although various hypotheses have been proposed to explain the rather unique physical properties of this asteroid, a perfect understanding of its formation and bulk composition is still missing. We aim to refine the shape and bulk density of (16) Psyche and to perform a thorough analysis of its shape to better constrain possible formation scenarios and the structure of its interior. We obtained disk-resolved VLT/SPHERE/ZIMPOL images acquired within our ESO large program (ID 199.C-0074), which complement similar data obtained in 2018. Both data sets offer a complete coverage of Psyche's surface. These images were used to reconstruct the three-dimensional (3D) shape of Psyche with two independent shape modeling algorithms (MPCD and ADAM). A shape analysis was subsequently performed, including a comparison with equilibrium figures and the identification of mass deficit regions. Our 3D shape along with existing mass estimates imply a density of 4.20+/-0.60g/cm^3, which is so far the highest for a solar system object following the four telluric planets. Furthermore, the shape of Psyche presents small deviations from an ellipsoid, that is, prominently three large depressions along its equator. The flatness and density of Psyche are compatible with a formation at hydrostatic equilibrium as a Jacobi ellipsoid with a shorter rotation period of ~3h. Later impacts may have slowed down Psyche's rotation, which is currently ~4.2h, while also creating the imaged depressions. Our results open the possibility that Psyche acquired its primordial shape either after a giant impact while its interior was already frozen or while its interior was still molten owing to the decay of the short-lived radionuclide ^26^Al.