The further exploration of the neural mechanisms underlying the biological activities of the human brain depends on the development of large-scale spiking neural networks (SNNs) with different categories at different levels, as well as the corresponding computing platforms. Neuromorphic engineering provides approaches to high-performance biologically plausible computational paradigms inspired by neural systems. In this article, we present a biological-inspired cognitive supercomputing system (BiCoSS) that integrates multiple granules (GRs) of SNNs to realize a hybrid compatible neuromorphic platform. A scalable hierarchical heterogeneous multicore architecture is presented, and a synergistic routing scheme for hybrid neural information is proposed. The BiCoSS system can accommodate different levels of GRs and biological plausibility of SNN models in an efficient and scalable manner. Over four million neurons can be realized on BiCoSS with a power efficiency of 2.8k larger than the GPU platform, and the average latency of BiCoSS is 3.62 and 2.49 times higher than conventional architectures of digital neuromorphic systems. For the verification, BiCoSS is used to replicate various biological cognitive activities, including motor learning, action selection, context-dependent learning, and movement disorders. Comprehensively considering the programmability, biological plausibility, learning capability, computational power, and scalability, BiCoSS is shown to outperform the alternative state-of-the-art works for large-scale SNN, while its real-time computational capability enables a wide range of potential applications.