We study bulge formation in MW/M31-like galaxies in a Λ-cold dark matter scenario, focusing on the origin of high- and low-Sersic index bulges. For this purpose, we use TNG50, a simulation of the IllustrisTNG project that combines a resolution of ~8 × 104 M⊙ in stellar particles with a cosmological volume 52 cMpc in extent. We parametrize bulge surface brightness profiles by the Sérsic index and the bulge-to-total (B/T) ratio obtained from two-component photometric decompositions. In our sample of 287 MW/M31-like simulated galaxies, $17.1{{\ \rm per\ cent}}$ of photometric bulges exhibit high-Sérsic indices and $82.9{{\ \rm per\ cent}}$ show low-Sérsic indices. We study the impact that the environment, mergers and bars have in shaping the surface brightness profiles. We find no correlation between bulge properties and the environment where they reside. Simulated galaxies with higher Sérsic indices show, on average, a higher fraction of ex situ stars in their kinematically selected bulges. For this bulge population, the last significant merger (total mass ratio msat/mhost > 0.1) occurs, on average, at later times. However, a substantial fraction of low-Sérsic index bulges also experience a late significant merger. We find that bars play an important role in the development of the different types of photometric bulges. The fraction of simulated galaxies with bars is smaller for the high- than for the low-Sérsic index population, reaching differences of $20{{\ \rm per\ cent}}$ at z > 1. Simulated galaxies with high fractions of ex situ stars in the bulge do not develop strong bars. Conversely, simulated galaxies with long-lived strong bars have bulges with ex situ fractions, fex situ < 0.2.