This paper presents an observer design technique for a newly developed non-intrusive position estimation system based on magnetic sensors. Typically, the magnetic field of an object as a function of position needs to be represented by a highly nonlinear measurement equation. Previous results on observer design for nonlinear systems have mostly assumed that the measurement equation is linear, even if the process dynamics are nonlinear. Hence, a new nonlinear observer design method for a Wiener system composed of a linear process model together with a nonlinear measurement equation is developed in this paper. First, the design of a two degree-of-freedom nonlinear observer is proposed that relies on a Lure system representation of the observer error dynamics. To improve the performance in the presence of parametric uncertainty in the measurement model, the nonlinear observer is augmented to estimate both the state and unknown parameters simultaneously. A rigorous nonlinear observability analysis is also presented to show that a dual sensor configuration is a sufficient and necessary condition for simultaneous state and parameter estimation. Finally, the developed observer design technique is applied to non-intrusive position estimation of the piston inside a pneumatic cylinder. Experimental results show that both position and unknown parameters can be reliably estimated in this application.