Monocular visual SLAM enables 3D reconstruction from internet video and autonomous navigation on resource-constrained platforms, yet suffers from scale drift, i.e., the gradual divergence of estimated scale over long sequences. Existing frame-to-frame methods achieve real-time performance through local optimization but accumulate scale drift due to the lack of global constraints among independent windows. To address this, we propose SCE-SLAM, an end-to-end SLAM system that maintains scale consistency through scene coordinate embeddings, which are learned patch-level representations encoding 3D geometric relationships under a canonical scale reference. The framework consists of two key modules: geometry-guided aggregation that leverages 3D spatial proximity to propagate scale information from historical observations through geometry-modulated attention, and scene coordinate bundle adjustment that anchors current estimates to the reference scale through explicit 3D coordinate constraints decoded from the scene coordinate embeddings. Experiments on KITTI, Waymo, and vKITTI demonstrate substantial improvements: our method reduces absolute trajectory error by 8.36m on KITTI compared to the best prior approach, while maintaining 36 FPS and achieving scale consistency across large-scale scenes.