<p>The Schrödinger equation is solved analytically for a particle subject to an enhanced Pöschl-Teller (EPT) potential in the presence of an external magnetic field and an Aharonov-Bohm flux field that induces topological phase effects. The bound state ro-vibrational energy equations are obtained using the generalized fractional Nikiforov-Uvarov method in conjunction with the Pekeris approximation scheme. The fractional order is introduced as an effective parameter that accounts for nonlocal and anharmonic ro-vibrational interactions that are not fully represented within the standard integer order framework. Based on these energy expressions, the mean thermal magnetization is derived within the partition function formalism. Numerical applications to diatomic molecules such as CO (X ¹Σ⁺), Cs₂ (3 ³Σ_g⁺), ICl (X ¹Σ_g⁺), ⁷Li₂ (1 ³Δ_g), Na₂ (C(2) ¹Π_u), and NaK (c ³Σ⁺) show that the mean percentage absolute deviation values decrease from 0.0990%, 0.1407%, 1.0027%, 1.9504%, 0.1234%, and 0.9811% to 0.0905%, 0.1020%, 0.5501%, 1.1756%, 0.0474%, and 0.4840% when fractional parameters are incorporated, indicating improved agreement with experimental data and enhanced flexibility of the ro-vibrational energy model. The analysis further shows that, at fixed temperatures, the mean thermal magnetization of the Na₂ (C(2) ¹Π_u) dimer increases with increasing magnetic field strength, highlighting the sensitivity of the system to external field variations. These results establish the EPT potential combined with a fractional order formulation as a reliable and adaptable analytical framework for describing the quantum and thermomagnetic behavior of diatomic molecules influenced by magnetic and topological quantum fields.</p>