Context. Circumstellar discs are essential for the formation of high mass stars, while multiplicity, and in particular binarity, appears to be an inevitable outcome, as the vast majority of massive stars (>8 M⊙) are found in binaries (up to 100%). Our understanding of the innermost regions of accretion discs around massive stars and the binarity of high-mass young stars is sparse because of the high spatial resolution and sensitivity required to trace these rare and distant objects. Aims. We aim to spatially resolve and constrain the sizes of the dust and ionised gas emission from the innermost regions of a sample of massive young stellar objects (MYSOs) for the first time, and to provide high-mass binary statistics for young stars at 2–300 au scales using direct interferometric measurements. Methods. We observed six MYSOs using long-baseline near-infrared K-band interferometry on the VLTI (GRAVITY, AMBER) in order to resolve and characterise the 2.2 μm hot dust emission originating from the inner rim of circumstellar discs around MYSOs, and the associated Brγ emission from ionised gas. We fitted simple geometrical models to the interferometric observables, and determined the inner radius of the dust emission. We placed MYSOs with K-band measurements in a size–luminosity diagram for the first time, and compared our findings to their low- and intermediate-mass counterparts (T Tauris and Herbig AeBes). We also compared the observed K-band sizes (i.e. inner rim radius) to the sublimation radius predicted by three different disc scenarios: a classical thick flattened structure with oblique heating in action, and direct heating from the protostar via an optically thin cavity with and without backwarming effects. Lastly, we applied binary geometries to trace close binarity among MYSOs. Results. The characteristic size of the 2.2 μm continuum emission towards this sample of MYSOs shows a large scatter at the given luminosity range. When the inner sizes of MYSOs are compared to those of lower mass Herbig AeBe and T Tauri stars, they appear to follow a universal trend in that the sizes scale with the square-root of the stellar luminosity. The Brγ emission originates from a similar or somewhat smaller and co-planar area compared to the 2.2 μm continuum emission. We discuss this new finding with respect to a disc-wind or jet origin. Finally, we report an MYSO binary fraction of 17–25% at milli-arcsecond separations (2–300 au). Conclusions. The size–luminosity diagram indicates that the inner regions of discs around young stars scale with luminosity independently of the stellar mass. The observed fraction of MYSO binaries in K-band is almost ‘flat’ for a wide range of separations (2–10 000 au). At the targeted scales (2–300 au), the MYSO binary fraction is lower than what was previously reported for the more evolved main sequence massive stars, which, if further confirmed, could implicate predictions from massive binary formation theories. Lastly, with this study, we can finally spatially resolve the crucial star–disc interface in a sample of MYSOs, showing that au-scale discs are prominent in high-mass star formation and are similar to their low-mass equivalents, while the ionised gas can be linked to disc wind and disc accretion models similar to Herbig AeBes.