We study the stellar mass function (SMF) of quiescent and star-forming galaxies and its dependence on morphology in ten redshift bins at 0.2 < z < 5.5. We used the COSMOS2025 catalog, built from the 0.54 deg2 JWST imaging from the COSMOS-Web survey, to select galaxies by type using the NUVrJ rest-frame color diagram and to classify them morphologically based on their bulge-to-total light ratio (B/T). The SMF of quiescent galaxies shows a rapid early build-up, with the most massive, log(M★/M⊙)≳11, being assembled by z ∼ 1 and evolving little since. The star-forming SMF evolves more slowly with redshift, following a mass-evolution scenario where galaxies grow in mass via star formation and quench once they reach the characteristic log(M∗/M⊙) ∼ 10.6. Bulge systems (B/T > 0.6) dominate the quiescent SMF at log(M★/M⊙) > 10 at all redshifts, while disk systems (B/T < 0.2) dominate at log(M★/M⊙) < 9. However, most bulge-dominated galaxies in the Universe are star forming, with their fraction increasing with redshift and decreasing mass, consistent with them being progenitors of quiescent bulges. We find evidence for the onset of environmental quenching as early as z ∼ 3 from the upturn in the quiescent SMF at log(M★/M⊙) < 9.5. This upturn is contributed by disk-dominated galaxies, consistent with environmental quenching scenarios in which satellites are quenched, but retain their disk morphologies. The number densities of log(M★/M⊙) > 10 quiescent galaxies are lower than in the recent literature by 0.1 − 0.7 dex, but agree well with cosmological galaxy formation simulations at 2 < z < 3. However, at z > 3, simulations increasingly underpredict the observations. Finally, we built a simple empirical model to describe the redshift evolution of galaxy number densities by parameterizing the quenching rate of all and bulge-dominated galaxies, stellar fraction, and bulge formation function. Our model is consistent with an evolutionary scenario where star-forming galaxies grow a central bulge before permanently quenching in massive halos.