HR\,2562\,B is a planetary-mass companion at an angular separation of $0.56 ($19$\,au) from the host star, which is also a member of a select number of L/T transitional objects orbiting a young star. This companion gives us a great opportunity to contextualize and understand the evolution of young objects in the L/T transition. However, the main physical properties (e.g., $ T_ eff $ and mass) of this companion have not been well constrained (34<!PCT!> uncertainties on $ T_ eff $, 22<!PCT!> uncertainty for log(g)) using only near-infrared (NIR) observations. We aim to narrow down some of its physical parameters uncertainties (e.g., $ T_ eff $: 1200K-1700K, log(g): 4-5) incorporating new observations in the Rayleigh-Jeans tail with the JWST/MIRI filters at $10.65$, $11.40$, and $15.50\ m$, as well as to understand its context in terms of the L/T transition and chemical composition. We processed the MIRI observations with reference star differential imaging (RDI) and detect the companion at high S/N (around $16$) in the three filters, allowing us to measure its flux and astrometry. We used two atmospheric models ATMO and Exo-REM to fit the spectral energy distribution using different combinations of mid-IR and near-IR datasets. We also studied the color-magnitude diagram using the F1065C and F1140C filters combined with field brown dwarfs to investigate the chemical composition in the atmosphere of HR\,2562\,B, as well as a qualitative comparison with the younger L/T transitional companion VHS\,1256\,b. We improved the precision on the temperature of HR\,2562\,B ($ T_ eff $ = $1255$\,K) by a factor of $6 compared to previous estimates ($ vs $ using ATMO . The precision of its luminosity was also narrowed down to $-4.69 dex. The surface gravity still presents a wider range of values (4.4 to 4.8 dex). While its mass was not narrowed down, we find the most probable values between $8 M_ Jup $ ($3$-sigma lower limit from our atmospheric modeling) and $18.5 M_ Jup $ (from the upper limit provided by astrometric studies). We report a sensitivity to objects of mass ranging between $2-5 M_ Jup $ at $100$\,au, reaching the lower limit at F1550C . We also implemented a few improvements in the pipeline related to the background subtraction and stages 1 and 2. HR\,2562\,B has a mostly (or near) cloud-free atmosphere, with the ATMO model demonstrating a better fit to the observations. From the color-magnitude diagram, the most probable chemical species at MIR wavelengths are silicates (but with a weak absorption feature); however, follow-up spectroscopic observations are necessary to either confirm or reject this finding. The mass of HR\,2562\,B could be better constrained with new observations at $3-4 m$. Although HR\,2562\,B and VHS\,1256\,b have very similar physical properties, both are in different evolutionary states in the L/T transition, which makes HR\,2562\,B an excellent candidate to complement our knowledge of young objects in this transition. Considering the actual range of possible masses, HR\,2562\,B could be considered as a planetary-mass companion; hence, its name then ought to be rephrased as HR\,2562\,b.
