Failure classification (FC) is a crucial problem in microservice systems, as it enables precise failure location, reduces the mean time to repair (MTTR), and ensures that service level agreements (SLAs) are maintained. However, existing methods for FC mostly rely on independent anomaly detectors or cascade feature extraction modules to handle multimodal monitoring data (e.g., logs, metrics, and traces), which suffer from error accumulation and amplification over multi-stage pipelines, leading to suboptimal performance. To address this issue, we propose FC-VGAE, a new failure classification method based on the variational graph auto-encoder with multimodal data fusion and joint feature extraction. Specifically, it first builds microservice invocation graphs (MIGs) from monitoring data. Then, it utilizes a semi-supervised VGAE to capture the normal behavior of the microservice system and produces the reconstruction errors for all nodes in MIGs, which are fed into a multi-layer perceptron (MLP) to classify the failure types. Finally, we evaluate FC-VGAE on two large-scale real-world microservice datasets. The results show that FC-VGAE improves over state-of-the-art baseline methods by about 21% and 19%, respectively, in F1-scores on the two datasets, validating its superiority for microservice failure classification.