The photometric catalogue of galaxy clusters extracted from ESA Euclid data is expected to be very competitive for cosmological studies. Using dedicated hydrodynamical simulations, we present systematic analyses simulating the expected weak-lensing profiles from clusters in a variety of dynamic states and for a wide range of redshifts. In order to derive cluster masses, we use a model consistent with the implementation within the Euclid Consortium of the dedicated processing function and find that when we jointly model the mass and concentration parameter of the Navarro-Frenk-White halo profile, the weak-lensing masses tend to be biased low by 5-10% on average with respect to the true mass, up to z = 0.5. For a fixed value for the concentration c200 = 3, the mass bias is decreases to lower than 5%, up to z = 0.7, along with the relative uncertainty. Simulating the weak-lensing signal by projecting along the directions of the axes of the moment of inertia tensor ellipsoid, we find that orientation matters: when clusters are oriented along the major axis, the lensing signal is boosted, and the recovered weak-lensing mass is correspondingly overestimated. Typically, the weak-lensing mass bias of individual clusters is modulated by the weak-lensing signal-to-noise ratio, which is related to the redshift evolution of the number of galaxies used for weak-lensing measurements: the negative mass bias tends to be stronger toward higher redshifts. However, when we use a fixed value of the concentration parameter, the redshift evolution trend is reduced. These results provide a solid basis for the weak-lensing mass calibration required by the cosmological application of future cluster surveys from Euclid and Rubin.