Autonomous racing provides a controlled environment for testing the software and hardware of autonomous vehicles operating at their performance limits. Competitive interactions between multiple autonomous racecars however introduce challenging and potentially dangerous scenarios. Accurate and consistent vehicle detection and tracking is crucial for overtaking maneuvers, and low-latency sensor processing is essential to respond quickly to hazardous situations. This paper presents the LiDAR-based perception algorithms deployed on Team PoliMOVE's autonomous racecar, which won multiple competitions in the Indy Autonomous Challenge series. Our Vehicle Detection and Tracking pipeline is composed of a novel fast Point Cloud Segmentation technique and a specific Vehicle Pose Estimation methodology, together with a variable-step Multi-Target Tracking algorithm. Experimental results demonstrate the algorithm's performance, robustness, computational efficiency, and suitability for autonomous racing applications, enabling fully autonomous overtaking maneuvers at velocities exceeding 275 km/h.