Physically based electrochemical models can provide internal state information of batteries, exhibiting significant potential for application in battery management systems across various lithium-ion battery products. However, the complex structure and limited observability of electrochemical models make their integration into battery management systems challenging. In this paper, a novel nonlinearity-aware adaptive (i.e., NAA) observer based on a simplified electrochemical model is proposed for estimating the electrode surface concentration and state of charge. The proposed NAA observer can identify the nonlinearity of the electrochemical model and switches to the most suitable observer structure automatically. Experimental tests on a cylindrical battery were conducted to observe the performance of the observer. The results demonstrate rapid convergence and high accuracy of all estimated states under an initial state error of 20%. Furthermore, to thoroughly validate the performance of the proposed observer, a comparison was made with an interconnected sliding mode observer based on electrochemical models, considering initial state uncertainty, sensor noise, and computation time.