Digital Twins (DT) have the potential to transform traffic management and operations by creating dynamic, virtual representations of transportation systems that sense conditions, analyze operations, and support decision-making. A key component for DT of the transportation system is dynamic roadway geometry sensing. However, existing approaches often rely on static maps or costly sensors, limiting scalability and adaptability. Additionally, large-scale DTs that collect and analyze data from multiple sources face challenges in privacy, communication, and computational efficiency. To address these challenges, we introduce Geo-ORBIT (Geometrical Operational Roadway Blueprint with Integrated Twin), a unified framework that combines real-time lane detection, DT synchronization, and federated meta-learning. At the core of Geo-ORBIT is GeoLane, a lightweight lane detection model that learns lane geometries from vehicle trajectory data using roadside cameras. We extend this model through Meta-GeoLane, which learns to personalize detection parameters for local entities, and FedMeta-GeoLane, a federated learning strategy that ensures scalable and privacy-preserving adaptation across roadside deployments. Our system is integrated with CARLA and SUMO to create a high-fidelity DT that renders highway scenarios and captures traffic flows in real-time. Extensive experiments across diverse urban scenes show that FedMeta-GeoLane consistently outperforms baseline and meta-learning approaches, achieving lower geometric error and stronger generalization to unseen locations while drastically reducing communication overhead. This work lays the foundation for flexible, context-aware infrastructure modeling in DTs. The framework is publicly available at https://github.com/raynbowy23/FedMeta-GeoLane.git.