Aims. We continue the analysis of the data set of our spectroscopic observation campaign of M 31, whose ultimate goal is to provide an understanding of the three-dimensional structure of the bulge, its formation history, and composition in terms of a classical bulge, boxy-peanut bulge, and bar contributions. Methods. We derive simple stellar population (SSP) properties, such as age metallicity and α-element overabundance, from the measurement of Lick/IDS absorption line indices. We describe their two-dimensional maps taking into account the dust distribution in M 31. Results. We found 80% of the values of our age measurements are larger than 10 Gyr. The central 100 arcsec of M 31 are dominated by the stars of the classical bulge of M 31. These stars are old (11−13 Gyr), metal-rich (as high as [Z/H] ≈ 0.35 dex) at the center with a negative gradient outward and enhanced in α-elements ([α/Fe]≈ 0.28±0.01 dex). The bar stands out in the metallicity map, where an almost solar value of [Z/H] (≈0.02 ± 0.01 dex) with no gradient is observed along the bar position angle (55.7 deg) out to 600 arcsec from the center. In contrast, no signature of the bar is seen in the age and [α/Fe] maps, which are approximately axisymmetric, delivering a mean age and overabundance for the bar and boxy-peanut bulge of 10–13 Gyr and 0.25–0.27 dex, respectively. The boxy-peanut bulge has almost solar metallicity (−0.04 ± 0.01 dex). The mass-to-light ratio of the three components is approximately constant at M/L_V ≈ 4.4−4.7 M_⊙/L_⊙. The disk component at larger distances is made of a mixture of stars, as young as 3–4 Gyr, with solar metallicity and smaller M/L_V (≈3 ± 0.1 M_⊙/L_⊙). Conclusions. We propose a two-phase formation scenario for the inner region of M 31, where most of the stars of the classical bulge come into place together with a proto-disk, where a bar develops and quickly transforms it into a boxy-peanut bulge. Star formation continues in the bulge region, producing stars younger than 10 Gyr, in particular along the bar, thereby enhancing its metallicity. The disk component appears to build up on longer timescales.