Characterizing exoplanet's spectra is a crucial step in understanding the chemical and physical processes shaping their atmospheres and constraining their formation and evolutionary history. The X-SHYNE library is a homogeneous sample of 43 medium-resolution (R~8000) infrared (0.3-2.5um) spectra of young (<500Myr), low-mass (<20M_jup_), and cold (Teff~600-2000K) isolated brown dwarfs and wide-separation companions observed with the VLT/X-Shooter instrument. To characterize our targets, we performed a global comparative analysis. We first applied a semi-empirical approach. By refining their age and bolometric luminosity, we derived key atmospheric and physical properties, such as Teff, mass, surface gravity (g), and radius, using the evolutionary model COND03. These results were then compared with the results from a synthetic analysis based on three self-consistent atmospheric models: the cloudy models Exo-REM and Sonora Diamondback, and the cloudless model ATMO. To compare our spectra with these grids we used the Bayesian inference code ForMoSA. We found similar Lbol estimates between both approaches, but an underestimated Teff from the cloudy models, likely due to a lack of absorbers that could dominate the J and H bands of early L. We also observed a discrepancy in the log(g) estimates, which are dispersed between 3.5 and 5.5dex for mid-L objects. We interpreted this as a bias caused by a range of rotational velocities leading to cloud migration toward equatorial latitudes, combined with a variety of viewing angles that result in different observed atmospheric properties (cloud column densities, atmospheric pressures, etc.). This interpretation is supported by the correlation of the color anomaly D(J-K) of each object with log(g) and the parameter fsed that drives the sedimentation of the clouds. Finally, while providing robust estimates of [M/H] and C/O for individual objects remains challenging, the X-SHYNE library globally suggests solar values, which are consistent with a formation via stellar formation mechanisms. This study highlights the strength of homogeneous datasets in performing comparative analyses, reducing the impact of systematics, and ensuring robust conclusions while avoiding over-interpretation.