The rapid increase of mobile users and advancement of widely used applications introduce high network demands for low-latency and reliable mobility management in mobile communication networks. However, the traditional handover approaches are rule-based and rely solely on signal strength thresholds with hysteresis margins, which are prone to ping-pong effects and are unable to adapt to dynamic network conditions. Machine Learning (ML) models have been integrated for handover predictions, but their centralized architecture compromises user data privacy, which conflicts with the General Data Protection Regulation (GDPR). These centralized ML approaches also introduce scalability constraints that limit their effectiveness in dense network deployments. To address these challenges, this work proposes a Federated Learning with Software-Defined Mobile Networking (FL-SDMN) framework, a unified approach that integrates federated privacy-preserving learning with centralized network coordination for intelligent handover optimization in 5G and beyond networks. The framework leverages a lightweight federated ExtraTrees ensemble model with weighted tree-based aggregation to preserve data privacy and SDMN to provide global network coordination. It has a three-layer decision pipeline that transforms handover control from a reactive threshold mechanism into a predictive, standards-aligned optimization process. Evaluation of the framework was done with real-world 5G mobility data in terms of decision latency, unnecessary handover reduction, and scalability across diverse network configurations. The findings indicate that the integration of FL, Extra Trees, and SDMN provides a scalable, privacy-preserving, and deployment-ready solution for intelligent mobility management in 5G and beyond networks.