The European Space Agency's Euclid mission will provide high-quality imaging for about 1.5 billion galaxies. A software pipeline to automatically process and analyse such a huge amount of data in real time is being developed by the Science Ground Segment of the Euclid Consortium; this pipeline will include a model-fitting algorithm, which will provide photometric and morphological estimates of paramount importance for the core science goals of the mission and for legacy science. The Euclid Morphology Challenge is a comparative investigation of the performance of five model-fitting software packages on simulated Euclid data, aimed at providing the baseline to identify the best-suited algorithm to be implemented in the pipeline. In this paper we describe the simulated dataset, and we discuss the photometry results. A companion paper is focussed on the structural and morphological estimates. We created mock Euclid images simulating five fields of view of 0.48 deg2 each in the IE band of the VIS instrument, containing a total of about one and a half million galaxies (of which 350 000 have a nominal signal-to-noise ratio above 5), each with three realisations of galaxy profiles (single and double Sérsic, and 'realistic' profiles obtained with a neural network); for one of the fields in the double Sérsic realisation, we also simulated images for the three near-infrared YE, JE, and HE bands of the NISP-P instrument, and five Rubin/LSST optical complementary bands (u, g, r, i, and z), which together form a typical dataset for an Euclid observation. The images were simulated at the expected Euclid Wide Survey depths. To analyse the results, we created diagnostic plots and defined metrics to take into account the completeness of the provided catalogues, as well as the median biases, dispersions, and outlier fractions of their measured flux distributions. Five model-fitting software packages (DeepLeGATo, Galapagos-2, Morfometryka, ProFit, and SourceXtractor++) were compared, all typically providing good results. Of the differences among them, some were at least partly due to the distinct strategies adopted to perform the measurements. In the best-case scenario, the median bias of the measured fluxes in the analytical profile realisations is below 1% at a signal-to-noise ratio above 5 in IE, and above 10 in all the other bands; the dispersion of the distribution is typically comparable to the theoretically expected one, with a small fraction of catastrophic outliers. However, we can expect that real observations will prove to be more demanding, since the results were found to be less accurate for the most realistic realisation. We conclude that existing model-fitting software can provide accurate photometric measurements on Euclid datasets. The results of the challenge are fully available and reproducible through an online plotting tool.