ABSTRACT In this second paper of the series, we present the two-dimensional (2D) emission line abundance analysis of NGC 628, the largest object within the PPAK Integral Field Spectroscopy (IFS) Nearby Galaxies Survey, PINGS. We introduce the methodology applied to the 2D IFS data in order to extract and deal with large spectral samples, from which a 2D abundance analysis can be later performed. We obtain the most complete and reliable abundance gradient of the galaxy up to date, by using the largest number of spectroscopic points sampled in the galaxy, and by comparing the statistical significance of different strong-line metallicity indicators. We find features not previously reported for this galaxy that imply a multimodality of the abundance gradient consistent with a nearly flat distribution in the innermost regions of the galaxy, a steep negative gradient along the disc and a shallow gradient or nearly constant metallicity beyond the optical edge of the galaxy. The N/O ratio seems to follow the same radial behaviour. We demonstrate that the observed dispersion in metallicity shows no systematic dependence with the spatial position, signal-to-noise ratio or ionization conditions, implying that the scatter in abundance for a given radius is reflecting a true spatial physical variation of the oxygen content. Furthermore, by exploiting the 2D IFS data, we were able to construct the 2D metallicity structure of the galaxy, detecting regions of metal enhancement and showing that they vary depending on the choice of the metallicity estimator. The analysis of axisymmetric variations in the disc of NGC 628 suggest that the physical conditions and the star formation history of different symmetric regions of the galaxy have evolved in a different manner. PUBLICATION ABSTRACT