In the past 15 years, the concentration-mass relation has been investigated diffusely in theoretical studies. On the other hand, only recently has this relation been derived from X-ray observations. When that happened, the results caused a certain level of concern: the X-ray normalizations and slopes were found significantly dissimilar from those predicted by theory. We analyzed 52 galaxy clusters and groups, simulated with different descriptions of the physical processes that affect the baryonic component, with the purpose of determining whether these discrepancies are real or induced by biases in the computation of the concentration parameter or in the determination of the selection function of the cluster sample for which the analysis is carried out. In particular, we investigate how the simulated concentration-mass relation depends (1) on the radial range used to derive the concentration; (2) on the presence of baryons in the simulations, and on the effect of star formation and feedback from supernovae and active galactic nuclei (AGNs). Finally, we evaluate (3) how the results differ when adopting an X-ray approach for the analysis and (4) how the selection function based on X-ray luminosity can impact the results. All effects studied go in the direction of alleviating the discrepancy between observations and simulations, although with different significance: while the choice of the radial range to fit the profiles and the inclusion of the baryonic component play only a minor role, the X-ray approach to reconstruct the mass profiles and the selection of the cluster sample have a strong impact on the resulting concentration-mass relation. Extending the fit to the most central regions or reducing the fitting radius from the virial boundary to the typical X-ray external radius causes an increase of the normalization in radiative simulations by 5%-10%. In the second case, we measure a slope that is up to twice steeper than that derived by using the typical theoretical radial range. Radiative simulations including only supernova feedback produce 30% higher concentrations than the dark matter case. Such a difference is largely reduced when including the effect of AGN feedback. The concentration-mass relation derived from the X-ray synthetic catalog is significantly steeper due to the combination of several different effects, such as environment, dynamical state and dynamical history of the clusters, bias in mass and temperature measurements, and their dependence on the radius and on the mass of the system. Finally, selecting clusters according to their X-ray luminosity produces a net increase in both normalization and slope of the relation, since at fixed mass, the most luminous clusters are also the most concentrated.