This work is based on high-quality integral-field spectroscopic data obtained with the Multi Unit Spectroscopic Explorer (MUSE) on the Very Large Telescope (VLT). The 21 brightest (mB ≤ 15 mag) early-type galaxies (ETGs) inside the virial radius of the Fornax cluster are observed out to distances of ∼2−3 Re. Deep imaging from the VLT Survey Telescope (VST) is also available for the sample ETGs. We investigated the variation of the galaxy structural properties as a function of the total stellar mass and cluster environment. Moreover, we correlated the size scales of the luminous components derived from a multi-component decomposition of the VST surface-brightness radial profiles of the sample ETGs with the MUSE radial profiles of stellar kinematic and population properties. The results are compared with both theoretical predictions and previous observational studies and used to address the assembly history of the massive ETGs of the Fornax cluster. We find that galaxies in the core and north-south clump of the cluster, which have the highest accreted mass fraction, show milder metallicity gradients in their outskirts than the galaxies infalling into the cluster. We also find a segregation in both age and metallicity between the galaxies belonging to the core and north-south clump and the infalling galaxies. The new findings fit well within the general framework for the assembly history of the Fornax cluster.