Context. Q 0151+048 is a physical quasar (QSO) pair at z ~ 1.929 with a separation of 3.3 arcsec on the sky. In the spectrum of the brighter member of this pair, Q 0151+048A, a damped Lyα absorber (DLA) is observed at a higher redshift. We have previously detected the host galaxies of both QSOs, as well as a Lyα blob whose emission surrounding Q 0151+048A extends over 5 × 3.3 arcsec. Aims: We seek to constrain the geometry of the system and understand the possible relations between the DLA, the Lyα blob, and the two QSOs. We also aim at characterizing the former two objects in more detail. Methods: To study the nature of the Lyα blob, we performed low-resolution, long-slit spectroscopy with the slit aligned with the extended emission. We also observed the whole system using the medium-resolution VLT/X-shooter spectrograph and the slit aligned with the two QSOs. The systemic redshift of both QSOs was determined from rest-frame optical emission lines redshifted into the NIR. We employed line-profile fitting technique, to measure metallicities and the velocity width of low-ionization metal absorption lines associated to the DLA and photo-ionization modeling to characterize the DLA further. Results: We measure systemic redshifts of zem(A) = 1.92924 ± 0.00036 and zem(B) = 1.92863 ± 0.00042 from the H β and H α emission lines, respectively. In other words, the two QSOs have identical redshifts within 2σ. From the width of Balmer emission lines and the strength of the rest-frame optical continuum, we estimate the masses of the black holes of the two QSOs to be 109.33 M&sun; and 108.38 M&sun; for Q 0151+048A and Q 0151+048B, respectively. We then use the correlation between black hole mass and dark matter halo mass to infer the mass of the dark matter halos hosting the two QSOs: 1013.74 M&sun; and 1013.13 M&sun; for Q 0151+048A and Q 0151+048B, respectively. We observe a velocity gradient along the major axis of the Lyα blob consistent with the rotation curve of a large disk galaxy, but it may also be caused by gas inflow or outflow. We detect residual continuum in the DLA trough, which we interpret as emission from the host galaxy of Q 0151+048A. The derived H0 column density of the DLA is log NH0 = 20.34 ± 0.02 cm-2. Metal column densities are also determined for a number of low-ionization species resulting in an overall metallicity of 0.01 Z&sun;. We detect C ii ∗ , which allows us to make a physical model of the DLA cloud. Conclusions: From the systemic redshifts of the QSOs, we conclude that the Lyα blob is associated with Q 0151+048A rather than with the DLA. The DLA must be located in front of both the Lyα blob and Q 0151+048A at a distance greater than 30 kpc and has a velocity relative to the blob of 640 ± 70 km s-1. The two quasars accrete at normal Eddington ratios. The DM halo of this double quasar will grow to the mass of our local supercluster at z = 0. We point out that those objects therefore form an ideal laboratory to study the physical interactions in a z = 2 precursor of our local supercluster. Based on observations done with i) European Southern Observatory (ESO) utilizing 8.2m Very Large Telescope (VLT) X-shooter spectrograph on Cerro Paranal in the Atacama Desert, northern Chile. ii) 2.56 m Nordic Optical Telescope (NOT), a scientific association between Denmark, Finland, Iceland, Norway and Sweden, operated at Observatorio del Roque de Los Muchachos on the island of La Palma, Spain.