A new robust and accurate approach for the detection and localization of flying objects with the purpose of highly dynamic aerial interception and agile multi-robot interaction is presented in this paper. The approach is proposed for use onboard an autonomous aerial vehicle equipped with a 3D Light Detection and Ranging (LiDAR) sensor providing input data for the algorithm. It relies on a novel 3D occupancy voxel mapping method for the target detection and a cluster-based multiple hypothesis tracker to compensate uncertainty of the sensory data. When compared to state-of-the-art methods of onboard detection of other flying objects, the presented approach provides superior localization accuracy and robustness to different environments and appearance changes of the target, as well as a greater detection range. Furthermore, in combination with the proposed multi-target tracker, sporadic false positives are suppressed, state estimation of the target is provided and the detection latency is negligible. This makes the detector suitable for tasks of agile multi-robot interaction, such as autonomous aerial interception or formation control where precise, robust, and fast relative localization of other robots is crucial. We demonstrate the practical usability and performance of the system in simulated and real-world experiments.