As the exploration of deep-sea resources continues, underwater actuators with conventional motors as the main building blocks can no longer meet the increasingly demanding needs. Inspired by bionics, researchers have started to work on underwater actuators with bionic structures. In this study, we designed and implemented a novel Fishtail-like Pneumatic Soft Actuator (FPSA). This innovative actuator configuration is inspired by the tail structure of Body and/or Caudal Fin (BCF) mode fish. The actuator's motion is achieved by controlling the expansion and contraction of the pneumatic soft muscles on both sides. And by constructing an experimental platform, we conducted an in-depth performance characterization, revealing the existence of a frequency-dependent nonlinear hysteresis characteristic of the FPSA. In order to accurately characterize this property, we built a dynamic model of the FPSA and successfully identified the uncertain parameters in the model by applying the nonlinear least squares method. The validation results show that the constructed model can accurately describe the nonlinear hysteresis characteristics of the FPSA. Finally, we successfully realized the high-precision trajectory tracking control of the endpoint of the FPSA using a PID controller. This result provides relevant ideas for the research of novel underwater bionic actuators.