Adaptive human-agent and agent-agent cooperation are becoming more and more critical in the research area of multi-agent reinforcement learning (MARL), where remarked progress has been made with the help of deep neural networks. However, many established algorithms can only perform well during the learning paradigm but exhibit poor generalization during cooperation with other unseen partners. The personality theory in cognitive psychology describes that humans can well handle the above cooperation challenge by predicting others' personalities first and then their complex actions. Inspired by this two-step psychology theory, we propose a biologically plausible mixture of personality (MoP) improved spiking actor network (SAN), whereby a determinantal point process is used to simulate the complex formation and integration of different types of personality in MoP, and dynamic and spiking neurons are incorporated into the SAN for the efficient reinforcement learning. The benchmark Overcooked task, containing a strong requirement for cooperative cooking, is selected to test the proposed MoP-SAN. The experimental results show that the MoP-SAN can achieve both high performances during not only the learning paradigm but also the generalization test (i.e., cooperation with other unseen agents) paradigm where most counterpart deep actor networks failed. Necessary ablation experiments and visualization analyses were conducted to explain why MoP and SAN are effective in multi-agent reinforcement learning scenarios while DNN performs poorly in the generalization test.