The integration of backscatter communication (BackCom) technology with integrated sensing and communication (ISAC) technology not only enhances the system sensing performance, but also enables low-power information transmission. This is expected to provide a new paradigm for communication and sensing in internet of everything (IoE) applications. In this paper, we propose a novel cognitive wireless system called backscatter-ISAC (B-ISAC) and develop a joint beamforming framework for different stages (task modes). This system can achieve cognitive spectrum sharing between legacy communication, backscatter communication and sensing functions. We derive communication performance metrics of the system in terms of the signal-to-interference-plus-noise ratio (SINR) and communication rate, and derive sensing performance metrics of the system in terms of probability of detection, error of linear least squares (LS) estimation, and the error of linear minimum mean square error (LMMSE) estimation. The proposed joint beamforming framework consists of three stages: tag detection, tag estimation, and communication enhancement. We develop corresponding joint beamforming schemes aimed at enhancing the performance objectives of their respective stages by solving complex non-convex optimization problems. Extensive simulation results demonstrate the effectiveness of the proposed joint beamforming schemes. The proposed B-ISAC system has broad application prospect in next generation IoE scenarios.