We present the results from VLT/X-shooter spectroscopic observations of 11 extremely strong intervening damped Lyman-α absorbers (ESDLAs) that were initially selected as high N(H I) (i.e. ≥5 × 10²¹ cm⁻²) candidates from the Sloan Digital Sky Survey (SDSS). We confirm the high H I column densities, which we measure to be in the range log N(H I) = 21.6 − 22.4. Molecular hydrogen is detected with high column densities (N(H₂)≥10¹⁸ cm⁻²) in 5 out of 11 systems, 3 of which are reported here for the first time, and we obtain conservative upper limits on N(H₂) for the remaining 6 systems. We also measure the column density of various metal species (Zn II, Fe II, Si II, Cr II, and C I), quantify the absorption-line kinematics (Δv₉₀), and estimate the extinction of the background quasar light (A_V) by dust in the absorbing gas. We compare the chemical properties of this sample of ESDLAs, supplemented with literature measurements, to that of DLAs located at the redshift of long-duration γ-ray bursts (GRB-DLAs). We confirm that the two populations are almost indistinguishable in terms of chemical enrichment and gas kinematics. In addition, we find no marked differences in the incidence of H₂. All this suggests that ESDLAs and GRB-DLAs probe similar galactic environments. We search for the galaxy counterparts of ESDLAs and find associated emission lines in 3 out of 11 systems, 2 of which are reported here for the first time (at z_abs = 2.304 and 2.323 towards the quasars SDSS J002503.03+114547.80 and SDSS J114347.21+142021.60, respectively). The measured separations between the quasar sightlines and the emission associated with the ESDLA galaxy (for a total of five sightlines) are all very small (ρ <  3 kpc). Because our observations are complete up to ρ ∼ 7 kpc, we argue that the emission counterparts of the remaining systems are more likely below the detection limit than outside the search area. While the small impact parameters are similar to what is observed for GRB-DLAs, the associated star formation rates are on average lower than for GRB host galaxies. This is explained by long-duration GRBs being associated with the death of massive stars and therefore pinpointing regions of active star formation in the GRB host galaxies. Our observations support the suggestion from the literature that ESDLAs could act as blind analogues of GRB-DLAs, probing neutral gas with high column density in the heart of high-redshift galaxies, without any prior on the instantaneous star formation rate.