Semantic communication has been introduced into collaborative perception systems for autonomous driving, offering a promising approach to enhancing data transmission efficiency and robustness. Despite its potential, existing semantic communication approaches predominantly rely on analog transmission models, rendering these systems fundamentally incompatible with the digital architecture of modern vehicle-to-everything (V2X) networks and posing a significant barrier to real-world deployment. To bridge this critical gap, we propose CoDS, a novel collaborative perception framework based on digital semantic communication, designed to realize semantic-level transmission efficiency within practical digital communication systems. Specifically, we develop a semantic compression codec that extracts and compresses task-oriented semantic features while preserving downstream perception accuracy. Building on this, we propose a novel semantic analog-to-digital converter that converts these continuous semantic features into a discrete bitstream, ensuring integration with existing digital communication pipelines. Furthermore, we develop an uncertainty-aware network (UAN) that assesses the reliability of each received feature and discards those corrupted by decoding failures, thereby mitigating the cliff effect of conventional channel coding schemes under low signal-to-noise ratio (SNR) conditions. Extensive experiments demonstrate that CoDS significantly outperforms existing semantic communication and traditional digital communication schemes, achieving state-of-the-art perception performance while ensuring compatibility with practical digital V2X systems.