Context. Theories of the formation and evolution of small bodies in planetary systems predict that they may escape into interstellar space at any time, leaked from extrasolar Oort clouds or ejected following close encounters with local planets or their host stars. After just two such interlopers – 1I/2017 U1 ('Oumuamua) and 2I/Borisov – had been characterized, more questions had been raised than answered. Assessing the recently discovered interstellar comet 3I/ATLAS will only broaden our understanding of this complex topic. Aims. Here, we investigate the spectral, cometary, and rotational properties of 3I/ATLAS as well as its dynamical context. Methods. We identified the spectral type of 3I/ATLAS from the visible reflectance spectrum and used photometric observations to derive its level of activity and rotational properties. Observational data were obtained with the OSIRIS camera spectrograph at the 10.4 m Gran Telescopio Canarias and the Two-meter Twin Telescope. We used N-body simulations and statistical analyses of Gaia DR3 data to investigate the origin of 3I/ATLAS and its Galactic background. Results. Interstellar comet 3I/ATLAS has a visible spectrum slightly redder than the ones of D-type asteroids, 1I/'Oumuamua and 2I/Borisov, with a spectral slope of S' = 18.3 ± 0.9 %/1000 Å in the 4000–9000 Å range, which is similar to the ones of trans-Neptunian objects and Centaurs. It has a conspicuous coma and its rotation period is 16.79 ± 0.23 h. The heliocentric components of its Galactic velocity are (U,V,W) = (‑51.233 ± 0.006, ‑19.456 ± 0.004, +18.930‑0.006+0.005) km s‑1 with a radiant in Sagittarius of (α, δ) = (295.°043‑0.°004+0.°003, ‑19.°0704‑0.°0005+0.°0006). The analysis of a sample of kinematic analogs of 3I/ATLAS extracted from Gaia DR3 suggests that its parent system is part of the Galactic thin disk and includes a solar-like star with a slightly subsolar metallicity. Conclusions. The results from the physical characterization of 3I/ATLAS further support the idea that extrasolar debris is not too different from the debris found in the Solar System and that it is the result of similar formation processes. ⋆ Based on observations made with the Gran Telescopio Canarias (GTC) telescope (program ID 62-GTC55/24B), in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias (IAC), and the Two-meter Twin Telescope (TTT, PEI project PLANETIX25), in the Spanish Observatorio del Teide of the IAC.