Observational signatures of accretion disc winds have been found in a significant number of low-mass X-ray binaries at either X-ray or optical wavelengths. The 2015 outburst of the black hole transient V404 Cygni provided a unique opportunity for studying both types of outflows in the same system. We used contemporaneous X-ray (Chandra Observatory) and optical (Gran Telescopio Canarias, GTC) spectroscopy, in addition to hard X-ray light curves (INTEGRAL). We show that the kinetic properties of the wind, as derived from P-Cyg profiles detected in the optical range at low hard X-ray fluxes and in a number of X-ray transitions during luminous flares, are remarkably similar. Furthermore, strictly simultaneous data taken at intermediate hard X-ray fluxes show consistent emission line properties between the optical and the X-ray emission lines, which most likely arise in the same accretion disc wind. We discuss several scenarios to explain the properties of the wind, favouring the presence of a dynamic, multi-phase outflow during the entire outburst of the system. This study, together with the growing number of wind detections with fairly similar characteristic velocities at different wavelengths, suggest that wind-type X-ray binary outflows might be predominantly multi-phase in nature.