Access to medium-resolution spectra (R_lambda_~1000-10000) at near-infrared wavelengths of young M-L objects allows us to study their atmospheric properties. Specifically, this approach can unveil a rich set of molecular features related to the atmospheric chemistry and physics. We aim to deepen our understanding of the M-L transition on planetary-mass companions and isolated brown dwarfs, while searching for evidence of possible differences between these two populations of objects. To this end, we present a set of 21 VLT/SINFONI K-band (1.95-2.45um) observations from five archival programs at R_lambda_~4000. We aim to measure the atmospheric properties, such as Teff, log(g), [M/H], and C/O, and to understand the similarities and differences between objects ranging in spectral type from M5 to L5. We extracted the spectra of these targets with the TExTRIS code. We modeled them using ForMoSA, a Bayesian forward modeling tool for spectral analysis, and we explored four families of self-consistent atmospheric models: ATMO, BT-Settl, Exo-REM, and Sonora Diamondback. Here, we present the spectra of our targets and the derived parameters from the atmospheric modeling process. We confirm a drop in Teff as a function of the spectral type of more than 500 K at the M/L transition. In addition, we report C/O measurements for three companions, 2M 0103 AB b, AB Pic b, and CD-35 2722 b, thereby adding to the growing list of exoplanets with measured C/O ratios. The VLT/SINFONI Library highlights two key points. First, there is a critical need to further investigate the discrepancies among grids of spectra generated by self-consistent models, as these models yield varying results and do not uniformly explore the parameter space. Second, we do not observe any obvious discrepancies in the K-band spectra between companions and isolated brown dwarfs, which suggests that these super-Jupiter objects might have formed through a similar process; however, this possibility warrants further investigation.