Context. The possible existence of a continuum encompassing the diversity of explosive stellar deaths, ranging from ordinary supernovae (SNe; lacking any sign of a relativistic outflow) to relativistic hypernovae associated with energetic long duration gamma-ray bursts (GRBs), is under intense debate. In this context, the supernova SN 2008D associated with the X-ray transient (XT) 080109 could represent a paradigmatic case, since it might exemplify a potential borderline transition event. Optical polarimetric studies could contribute to shed light on the different interpretations given in the literature for this supernova (hereafter, SN 2008D/XT 080109). Aims: The main aim is to infer geometric information of SN 2008D/XT 080109 through the study of the evolution of its linear optical polarization. We also report the polarization evolution of SN 2007uy, and discuss the properties of the host galaxy interstellar medium (ISM) towards the XT. The final goal is to compare the polarization properties, and therefore the geometries, of both SNe. Methods: We present a V-band linear polarization monitoring campaign carried out for SN 2008D/XT 080109 and SN 2007uy, which shone for weeks contemporaneously in NGC 2770. This fortunate coincidence brought us the opportunity to observe both objects simultaneously, and most importantly, with identical instrumental setups. The observations span 74.9 days, starting 3.6 days after the XT and are distributed in 11 visits. In addition we performed observations in the millimetre (mm) range in order to identify the dominant origin of the observed polarization. Results: We report positive linear polarization detections at several epochs for SN 2008D/XT 080109 at a level of ~1%. For SN 2007uy the measured polarization is around ~1.5%. In both cases the observed linear polarization seems dominated by the host galaxy interstellar polarization (HGIP), especially for the case of SN 2007uy. SN 2007uy shows Stokes parametres consistent with no time evolution, which could be described by the HGIP plus a constant eccentricity expansion on the sky plane. Over the course of our observations of SN 2007uy, we find that its total polarization signal does not change by more than 0.29% with a 90% confidence interval. Despite the dominant HGIP, a statistical analysis of the distribution of the SN 2008D/XT 080109 Stokes parametres suggests that it could show a possible intrinsic variable polarization component. Moreover, assuming the polarization signal from SN 2007uy is constant, we find that the temporal evolution of the intrinsic SN 2008D/XT 080109 polarization could be explained by an aspherical axisymmetric expansion with variable eccentricity, although other more complex geometric scenarios are also compatible with the data. We come to the same result even if we make no assumption on the SN 2007uy Stokes parametres, although at a lower significance level. Conclusions: We conclude that the data seem to suggest a potential symmetry axis for SN 2008D/XT 080109, which is reinforced when SN 2007uy is assumed to have constant Stokes parametres and used as reference star. We suggest that at least the projected, if not the intrinsic, geometry of SN 2008D/XT 080109 and SN 2007uy could be different.