The ability to measure unbiased weak-lensing (WL) masses is a key ingredient to exploit galaxy clusters as a competitive cosmological probe with the ESA Euclid survey or future missions. We investigate the level of accuracy and precision of cluster masses measured with the Euclid data processing pipeline. We use the DEMNUni-Cov N-body simulations to assess how well the WL mass probes the true halo mass, and, then, how well WL masses can be recovered in the presence of measurement uncertainties. We consider different halo mass density models, priors, and mass point estimates, that is the biweight, mean, and median of the marginalised posterior distribution and the maximum likelihood parameter. WL mass differs from true mass due to, for example, the intrinsic ellipticity of sources, correlated or uncorrelated matter and large-scale structure, halo triaxiality and orientation, and merging or irregular morphology. In an ideal scenario without observational or measurement errors, the maximum likelihood estimator is the most accurate, with WL masses biased low by {bM} =a-14.6-±-1.7% on average over the full range M200c > 5×1013 M⊙ and z < 1. Due to the stabilising effect of the prior, the biweight, mean, and median estimates are more precise, that is with smaller intrinsic scatter. The scatter decreases with increasing mass and informative priors can significantly reduce the scatter. Halo mass density profiles with a truncation provide better fits to the lensing signal, while the accuracy and precision are not significantly affected. We further investigate the impact of various additional sources of systematic uncertainty on the WL mass estimates, namely the impact of photometric redshift uncertainties and source selection, the expected performance of Euclid cluster detection algorithms, and the presence of masks. Taken in isolation, we find that the largest effect is induced by non-conservative source selection with {bM} =a-33.4-±-1.6%. This effect can be mostly removed with a robust selection. As a final Euclid-like test, we combine systematic effects in a realistic observational setting and find {bM} =a-15.5-±-2.4% under a robust selection. This is very similar to the ideal case, though with a slightly larger scatter mostly due to cluster redshift uncertainty and miscentering.