Identifying security patches via code commits to allow early warnings and timely fixes for Open Source Software (OSS) has received increasing attention. However, the existing detection methods can only identify the presence of a patch (i.e., a binary classification) but fail to pinpoint the vulnerability type. In this work, we take the first step to categorize the security patches into fine-grained vulnerability types. Specifically, we use the Common Weakness Enumeration (CWE) as the label and perform fine-grained classification using categories at the third level of the CWE tree. We first formulate the task as a Hierarchical Multi-label Classification (HMC) problem, i.e., inferring a path (a sequence of CWE nodes) from the root of the CWE tree to the node at the target depth. We then propose an approach named TreeVul with a hierarchical and chained architecture, which manages to utilize the structure information of the CWE tree as prior knowledge of the classification task. We collect a large security patch dataset from NVD, consisting of 6,541 commits from 1,560 GitHub OSS repositories. Experimental results show that TreeVul significantly outperforms the best performing baselines, with improvements of 5.9%, 25.0% and 7.7% in terms of weighted F1-score, macro F1-score and MCC, respectively. We further conduct a user study and a case study to verify the practical value of TreeVul in enriching the binary patch detection results and improving the data quality of NVD, respectively.