This paper proposes a novel online and self-learning algorithm to the identification of fuzzy neural networks, which not only learns the structure and parameters online but also learns the threshold parameters by itself and automatically. For structure learning, a self-constructing approach including adding neurons and merging highly similar fuzzy rules is proposed based on the criteria of the system error between actual and model output and the maximum firing strength of neurons. In order to achieve the efficient merging computing, a new calculation method of similarity degree between fuzzy rules is developed. Further and more importantly, the varying width of Gaussian membership functions can be learned by itself according to the underfitting and overfitting criteria. Similarly, different from the existing constant threshold of similarity degree for merging, the varying threshold of similarity degree can be self-learned according to the real-time accuracy of model. The proposed self-learning mechanism significantly improves the model accuracy and greatly enhances the easy usability. Several benchmark examples are implemented to illustrate the effectiveness and feasible of the proposed approach.