Thanks to the comprehensive sets of elemental abundances in eleven damped Ly$\alpha$ systems (DLAs) at $z_{\rm abs}=1.7-2.5$, we were able for the first time to study in detail the abundance patterns of a wide range of elements, the chemical variations in the interstellar medium, the star formation, and the age of galaxies outside the Local Group. Comparing the gas-phase abundance ratios of these high redshift galaxies, we found that they are very uniform, showing low RMS dispersions up to only 2–3 times higher than the statistical errors, for the majority of elements. The gas-phase abundance patterns of interstellar medium clouds within the DLA galaxies detected along the velocity profiles show on the other hand a high dispersion in several abundance ratios, indicating that variations in the gas phase, whatever their origin, are more confined to clouds within the DLA galaxies than to integrated profiles. The analysis of the cloud-to-cloud chemical variations within seven individual DLAs revealed that five of them show statistically significant variations, higher than 0.2 dex at more than 3 $\sigma$. The sources of these variations are both differential dust depletion and ionisation effects, however no evidence for variations due to different star formation histories was highlighted. These results place important constraints on scenarios of galaxy formation within the CDM hierarchical theory. The DLA abundance pattern comparison with chemical evolution models indicate that the DLAs may either be associated with the outer regions of spiral galaxies or with dwarf irregular galaxies both characterised by star formation histories with low star formation efficiencies. They may be very young galaxies with ages between 20–250 Myr, but also galaxies with ages longer than 1 Gyr. Their star formation rates per unit area are moderate, between $-3.2<\log {\rm SFR}<-1.4$ M$_{\odot}$ yr$^{-1}$ kpc$^{-2}$.