HR 2562 B is a planetary-mass companion at an angular separation of 0.56arcsec (19au) 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., Teff and mass) of this companion have not been well constrained (34% uncertainties on Teff , 22% uncertainty for log(g)) using only near-infrared (NIR) observations. We aim to narrow down some of its physical parameters uncertainties (e.g., Teff : 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 15.50um, 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 (Teff=1255K) by a factor of 6x compared to previous estimates (+/-15K vs +/-100K) using ATMO. The precision of its luminosity was also narrowed down to ~4.69+/-0.01dex. The surface gravity still presents a wider range of values (4.4 to 4.8dex). While its mass was not narrowed down, we find the most probable values between 8M_Jup_ (3{sigma} lower limit from our atmospheric modeling) and 18.5MJup (from the upper limit provided by astrometric studies). We report a sensitivity to objects of mass ranging between 2-5M_Jup_ at 100au, 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 3-4um. 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.