Trans-Neptunian objects (TNOs) are leftovers of planetary formation that orbit in the region beyond Neptune. Trans-Neptunian binaries (TNBs) provide a valuable opportunity to test the models of formation and evolution of planetesimals in the trans-Neptunian region. Some theories have been proposed to describe the observed separations between binary components, their relative sizes, and other orbital parameters. However, spectral information for the components of small TNBs, a crucial piece of information needed to understand formation mechanisms, has heretofore been unavailable in the literature. Here we present the first resolved spectra of a TNB, the Plutino (341520) Mors-Somnus, the only TNB with resolved components in the Large Cycle 1 James Webb Space Telescope (JWST) General Observer (GO) program "DiSCo-TNOs" (PID 2418; PI: Pinilla-Alonso). We use these spectra to gather information about the surface composition of the two components that can be used as an indicator of their dynamical evolution in the outer solar system. The spectra of Mors and Somnus are very similar to each other and are compatible with the surface composition of TNOs (using DiSCo data), in particular the Cold classical, presenting spectral features compatible with complex organic materials, and carbon dioxide. The similarity between Mors' and Somnus' spectra supports the streaming instability formation scenario for the pair, since they should have formed in the same region of the protoplanetary disk. Together with their dynamical history, the wide separation between the pair, the nearly equal size of the components, and the system's high orbital inclination suggest binary is a survivor of the primordial population of objects beyond 30 AU, where the Cold classical formed. The similarities found between the spectral features of Mors-Somnus and the Cold classical TNOs in the DiSCo-TNOs sample provide the first spectroscopic evidence for this scenario.