  The various Euclid imaging surveys will become a reference for studies of
galaxy morphology by delivering imaging over an unprecedented area of 15 000
square degrees with high spatial resolution. In order to understand the
capabilities of measuring morphologies from Euclid-detected galaxies and to
help implement measurements in the pipeline, we have conducted the Euclid
Morphology Challenge, which we present in two papers. While the companion paper
by Merlin et al. focuses on the analysis of photometry, this paper assesses the
accuracy of the parametric galaxy morphology measurements in imaging predicted
from within the Euclid Wide Survey. We evaluate the performance of five
state-of-the-art surface-brightness-fitting codes DeepLeGATo, Galapagos-2,
Morfometryka, Profit and SourceXtractor++ on a sample of about 1.5 million
simulated galaxies resembling reduced observations with the Euclid VIS and NIR
instruments. The simulations include analytic S\'ersic profiles with one and
two components, as well as more realistic galaxies generated with neural
networks. We find that, despite some code-specific differences, all methods
tend to achieve reliable structural measurements (10% scatter on ideal S\'ersic
simulations) down to an apparent magnitude of about 23 in one component and 21
in two components, which correspond to a signal-to-noise ratio of approximately
1 and 5 respectively. We also show that when tested on non-analytic profiles,
the results are typically degraded by a factor of 3, driven by systematics. We
conclude that the Euclid official Data Releases will deliver robust structural
parameters for at least 400 million galaxies in the Euclid Wide Survey by the
end of the mission. We find that a key factor for explaining the different
behaviour of the codes at the faint end is the set of adopted priors for the
various structural parameters.
