In this paper, we investigate the existence and uniqueness of global solutions to the Cauchy problem for a coupled stochastic chemotaxis-Navier-Stokes system with multiplicative Lévy noises in $\mathbb{R}^2$. The existence of global martingale solutions is proved under a framework that is based on the Faedo-Galerkin approximation scheme and stochastic compactness method, where the verification of tightness depends crucially on a novel stochastic version of Lyapunov functional inequality and proper compactness criteria in Fréchet spaces. A pathwise uniqueness result is also established with suitable assumption on the jump noises, which indicates that the considered system admits a unique global strong solution.