Surgical video segmentation is fundamental to computer-assisted surgery. In practice, surgeons need to dynamically specify targets throughout extended procedures, using heterogeneous cues such as visual selections, textual expressions, or audio instructions. However, existing Promptable Video Object Segmentation (PVOS) methods are typically restricted to a single prompt modality and rely on coupled frameworks that cause optimization interference between target initialization and tracking. Moreover, these methods produce hallucinated predictions when the target is absent and suffer from accumulated mask drift without failure recovery. To address these challenges, we present UniSurgSAM, a unified PVOS model enabling reliable surgical video segmentation through visual, textual, or audio prompts. Specifically, UniSurgSAM employs a decoupled two-stage framework that independently optimizes initialization and tracking to resolve the optimization interference. Within this framework, we introduce three key designs for reliability: presence-aware decoding that models target absence to suppress hallucinations; boundary-aware long-term tracking that prevents mask drift over extended sequences; and adaptive state transition that closes the loop between stages for failure recovery. Furthermore, we establish a multi-modal and multi-granular benchmark from four public surgical datasets with precise instance-level masklets. Extensive experiments demonstrate that UniSurgSAM achieves state-of-the-art performance in real time across all prompt modalities and granularities, providing a practical foundation for computer-assisted surgery. Code and datasets will be available at https://jinlab-imvr.github.io/UniSurgSAM.