This work presents EndoStreamDepth, a monocular depth estimation framework for endoscopic video streams. It provides accurate depth maps with sharp anatomical boundaries for each frame, temporally consistent predictions across frames, and real-time throughput. Unlike prior work that uses batched inputs, EndoStreamDepth processes individual frames with a temporal module to propagate inter-frame information. The framework contains three main components: (1) a single-frame depth network with endoscopy-specific transformation to produce accurate depth maps, (2) multi-level Mamba temporal modules that leverage inter-frame information to improve accuracy and stabilize predictions, and (3) a hierarchical design with comprehensive multi-scale supervision, where complementary loss terms jointly improve local boundary sharpness and global geometric consistency. We conduct comprehensive evaluations on two publicly available colonoscopy depth estimation datasets. Compared to state-of-the-art monocular depth estimation methods, EndoStreamDepth substantially improves performance, and it produces depth maps with sharp, anatomically aligned boundaries, which are essential to support downstream tasks such as automation for robotic surgery. The code is publicly available at https://github.com/MedICL-VU/EndoStreamDepth