Context. Trans-Neptunian objects (TNOs) are remnants of small icy bodies from planetary formation that orbit in the region beyond Neptune. Within the population of TNOs, Trans-Neptunian binaries (TNBs) provide a valuable opportunity to test the models of the formation and evolution of planetesimals in the trans-Neptunian region. Various theories have been proposed to describe the observed separations between binary components, their relative sizes, and other orbital parameters. The colors of TNOs have been used to trace the dynamical history of the outer Solar System and the colors of TNB components provide tests for formation theories. However, spectral information for the components of small TNBs, crucial information that could validate formation mechanisms, has until now remained elusive. Aims. The main goal of this work is to characterize the near-infrared spectral properties of the TNB plutino (341520) Mors–Somnus, the only TNB with resolved components in the James Webb Space Telescope (JWST) Large Cycle 1 General Observer program “DiSCo-TNOs” (PID 2418; PI: Pinilla-Alonso). The secondary goal is to use the surface compositions of the individual components of the Mors–Somnus system to probe formation and dynamical evolution in the outer Solar System through comparison to the surface properties of the cold classical and plutino (3:2 resonant) dynamical groups. Methods. To achieve these goals, we measured the spectral slope of the continuum and identified absorption bands in the individual spectra of Mors and Somnus, as well as in those of the cold classicals and plutinos obtained with the NIRSpec Integral Field Unit (IFU) and the PRISM/CLEAR disperser (0.6–5.3 μm), and compared these results to shed light on the dynamical evolution of the Mors–Somnus binary. Results. The spectra of Mors and Somnus are similar and indicate the presence of complex organic materials, CO 2 , CO, OH-compounds, and tentative nitrogen-rich materials. We find a high degree of compositional diversity in the plutino population, a group of TNOs that likely formed elsewhere and moved to their current orbits during the migration of Neptune, while the cold classical TNOs, which likely formed in situ, appear more homogeneous. Conclusions. The very wide separation between the components, their nearly equal sizes, and the high orbital inclination of the system suggest this plutino binary is a survivor of the primordial population of objects beyond 30 au. The similarities found between the spectral features of the plutinos Mors and Somnus and those of all of the cold classical TNOs in the DiSCo-TNOs sample as well as the high degree of compositional heterogeneity found in the plutino population provide compositional evidence for evaluation of Neptune’s migration in the trans-Neptunian region early on in the history of the Solar System.