Context. Morphological parameters are the estimators for the dynamical state of clusters of galaxies. Surveys performed at different wavelengths through their selection effects may be biased toward, different populations of clusters. For example, X-ray surveys are biased to detecting cool-core clusters as opposed to Sunyaev-Zel’dovich (SZ) surveys being more biased toward non-cool-core systems. Understanding the underlying population of clusters of galaxies in surveys is of the utmost importance when these samples are to be used in astrophysical and cosmological studies.Aims. We present an in-depth analysis of the X-ray morphological parameters of the galaxy clusters and groups detected in the eROSITA Final Equatorial-Depth Survey (eFEDS). The eFEDS, completed during the performance verification phase of the Spectrum-Roentgen-Gamma(SRG)/eROSITA telescope, is designed to provide the first eROSITA X-ray selected sample of galaxy clusters and groups.Methods. We studied the eROSITA X-ray imaging data for a sample of 325 clusters and groups that were significantly detected in the eFEDS field. We characterized their dynamical properties by measuring a number of dynamical estimators: concentration, central density, cuspiness, centroid shift, ellipticity, power ratios, photon asymmetry, and the Gini coefficient. The galaxy clusters and groups detected in eFEDS cover a luminosity range of more than three orders of magnitude and a wide redshift range out to 1.2. They provide an ideal sample on which the redshift and luminosity evolution of the morphological parameters can be studied and the underlying dynamical state of the sample can be characterized. Based on these measurements, we constructed a new dynamical indicator, the relaxation score, for all the clusters in the sample.Results. We find no evidence for a bimodality in the distribution of the morphological parameters of our clusters. We instead observe a smooth transition from the cool core to non-cool core and from relaxed to disturbed states, with a preference for skewed distributions or log-normal distributions. A significant evolution in redshift and luminosity is also observed in the morphological parameters we examined after taking the selection effects into account.Conclusions. We determine that in contrast do ROSAT-based cluster samples, our eFEDS-selected cluster sample is not biased toward cool-core clusters, but contains a similar fraction of cool-cores as SZ surveys.Key words: galaxies: clusters: intracluster medium / galaxies: clusters: general / X-rays: galaxies: clusters