Most dogs worldwide are free-ranging animals that form relationships mainly with conspecifics, yet research has focused mainly on the dog-human bond, leading to the hypothesis that dogs evolved specific abilities to form a unique relationship with humans. Although widespread, this hypothesis has not, as yet, been tested. Here we compared the relationships pet dogs form with their owner and with other dogs living in the same household. Using a bottom-up approach, we analyzed dogs' behavior in a test battery with both dog and human partners. Results revealed that pet dogs' relationships are characterized by three components (i.e. reference, affiliation and stress). A comparison between dogs' intra- and inter-specific relationships found that overall dogs refer more to their owner, but also that some dogs form stronger affiliative bonds with conspecifics than with their owner. Moreover, we tested how different partners could help dogs cope with a stressful situation. We found that the type of relationship, rather than the partner species, predicts how dogs react to a social threat. Our results suggest that dogs can form relationships of comparable qualities with both humans and other dogs, and that these relationships vary along multiple components across different partners.