We study the growth pathways of brightest central galaxies (BCGs) and intracluster light (ICE) by means of a semi-analytic model. We assume that ICL forms by stellar stripping of satellite galaxies and violent processes during mergers, and implement two independent models: One considers both mergers and stellar stripping (named STANDARD model), and the other considers only mergers (named MERGERS model). We find that BCGs and ICL form, grow, and overall evolve at different times and with different time-scales, but they show a clear coevolution after redshift z similar to 0.7-0.8. Around 90 per cent of the ICE from stellar stripping is built up in the innermost 150 kpc from the halo centre and the dominant contribution comes from disc-like galaxies (B/T < 0.4) through a large number of small/intermediate stripping events (M-strip/M-sat < 0.3). The fractions of stellar mass in BCGs and in the ICL over the total stellar mass within the virial radius of the halo evolve differently with time. At high redshift, the BCG accounts for the bulk of the mass, but its contribution gradually decreases with time and stays constant after z similar to 0.4-0.5. The ICL, instead, grows very fast and its contribution keeps increasing down to the present time. The STANDARD and MERGERS models make very similar predictions in most of the cases, but predict different amounts of ICL associated with galaxies within the virial radius of the group/cluster other than the BCG, at z = 0. We then suggest that this quantity is a valid observable that can shed light on the relative importance of mergers and stellar stripping for the formation of ICL.