Aims. We study the two main constituent galaxies of a constrained simulation of the Local Group as candidates for the Milky Way (MW) and Andromeda (M31). We focus on the formation of the stellar discs and its relation to the formation of the group as a rich system with two massive galaxies, and investigate the effects of mergers and accretion as drivers of morphological transformations. We also assess the effects of varying the assumed feedback model on our results by running two different simulations, a first one where only supernova feedback is included and a second where we additionally model radiation pressure from stars. Methods. We use a state-of-the-art hydrodynamical code which includes star formation, feedback and chemical enrichment to carry out our study. We use our two simulations, where we include or neglect the effects of radiation pressure from stars, to investigate the impact of this process on the morphologies and star formation rates of the simulated galaxies. Results. We find that the simulated M31 and MW have different formation histories, even though both inhabit, at z = 0, the same environment. These differences directly translate into and explain variations in their star formation rates, in-situ fractions and final morphologies. The simulated M31 candidate has an active merger history, as a result of which its stellar disc is unable to survive unaffected until the present time. In contrast, the MW candidate has a smoother history with no major mergers at late times, and forms a disc that grows steadily; at z = 0 the simulated MW has an extended, rotationally-supported disc which is dominant over the bulge. Our two feedback implementations predict similar evolution of the galaxies and their discs, although some variations are detected, the most important of which is the formation time of the discs: in the model with weaker/stronger feedback the discs form earlier/later. In summary, by comparing the formation histories of the two simulated galaxies, we conclude that the particular merger/accretion history of a galaxy rather than its environment at the LG-scales is the main driver of the formation and subsequent growth or destruction of galaxy discs.