We consider the problem of estimating the output of an unknown discrete-time linear time-invariant system and identifying a model of the system, where only measurements via a nonlinear dynamic sensor with known dynamics are available. The main result of this paper is a rank-constrained semidefinite program, which provides an equivalent characterization of this identification and estimation problem. This extends existing results from Wiener system identification to the more general case that the nonlinear block exhibits dynamic behavior, which is a commonly found scenario in practical applications. Notably, the result can be applied in the presence of nonlinear sensors with general non-invertible system dynamics. Two examples are used to illustrate the applicability of our approach.