We explore to which extent stars within Galactic disk open clusters resemble each other in the high-dimensional space of their photospheric element abundances, and contrast this with pairs of field stars. Our analysis is based on abundances for 20 elements, homogeneously derived from APOGEE spectra (with carefully quantified uncertainties, with a median value of $sim 0.03$ dex). We consider 90 red giant stars in seven open clusters and find that most stars within a cluster have abundances in most elements that are indistinguishable (in a $chi^2$-sense) from those of the other members, as expected for stellar birth siblings. An analogous analysis among pairs of $>1000$ field stars shows that highly significant abundance differences in the 20-dimensional space can be established for the vast majority of these pairs, and that the APOGEE-based abundance measurements have high discriminating power. However, pairs of field stars whose abundances are indistinguishable even at 0.03~dex precision exist: $sim 0.3$ percent of all field star pairs, and $sim 1.0$ percent of field star pairs at the same (solar) metallicity [Fe/H] = $0 pm 0.02$. Most of these pairs are presumably not birth siblings from the same cluster, but rather doppelganger. Our analysis implies that chemical tagging in the strict sense, identifying birth siblings for typical disk stars through their abundance similarity alone, will not work with such data. However, our approach shows that abundances have extremely valuable information for probabilistic chemo-orbital modeling and combined with velocities, we have identified new cluster members from the field.
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