Accurate differentiation between damage-related anomalies and data errors is a critical challenge in bridge monitoring. This paper presents a data-driven framework for anomaly detection and classification, addressing the question: How can anomalies be classified in multi-sensor bridge monitoring to distinguish structural changes from noise? The framework combines an adapted anomaly taxonomy with a deep neural network trained on synthetic data. It is validated using long-term monitoring data from a railway bridge, incorporating strain gauges, accelerometers, and an inclinometer. In offline training, the model achieves high precision, recall, and F1-scores, effectively detecting anomaly classes across sensor types. For online prediction, it provides anomaly type percentages and visualizations over daily, weekly, and annual timeframes, distinguishing frequent noise-related anomalies from rare anomalies signaling structural changes. Requiring one month of training data, the framework delivers a scalable solution for bridge monitoring and lays the groundwork for future self-learning anomaly detection in infrastructure management.