Using our sample of the most metal-rich damped Lyman $alpha$ systems (DLAs) at z$sim2$, and two literature compilations of chemical abundances in 341 DLAs and 2818 stars, we present an analysis of the chemical composition of DLAs in the context of the Local Group. The metal-rich sample of DLAs at z$sim2$ probes metallicities as high as the Galactic disc and the most metal-rich dwarf spheroidals (dSphs), permitting an analysis of many elements typically observed in DLAs (Fe, Zn, Cr, Mn, Si, and S) in comparison to stellar abundances observed in the Galaxy and its satellites (in particular dSphs). Our main conclusions are: (1) non-solar [Zn/Fe] abundances in metal-poor Galactic stars and in dSphs over the full metallicity range probed by DLAs, suggest that Zn is not a simple proxy for Fe in DLAs and therefore not a suitable indicator of dust depletion. After correcting for dust depletion, the majority of DLAs have subsolar [Zn/Fe] similar to dSphs; (2) at [Fe/H]$sim-0.5$, a constant [Mn/Fe]$sim-0.5$ and near-solar [$alpha$/Fe] (requiring an assumption about dust depletion) are in better agreement with dwarf galaxies than Galactic disc stars; (3) [$alpha$/Zn] is usually solar or subsolar in DLAs. However, although low ratios of [$alpha$/Fe] are usually considered more `dwarf-like than `Milky Way-like, subsolar [Zn/Fe] in Local Group dwarfs leads to supersolar [$alpha$/Zn] in the dSphs, in contrast with the DLAs. Therefore, whilst DLAs exhibit some similarities with the Local Group dwarf population, there are also notable differences.
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