The isomorphism problem, crucial in network analysis, involves analyzing both low-order and high-order structural information. Graph isomorphism algorithms focus on structural equivalence to simplify solver space, aiding applications like protein design, chemical pathways, and community detection. However, they fall short in capturing complex high-order relationships, unlike hypergraph isomorphism methods. Traditional hypergraph methods face challenges like high memory use and inaccurate identification, leading to poor performance. To overcome these, we introduce a hypergraph Weisfeiler-Lehman (WL) test algorithm, extending the WL test from graphs to hypergraphs, and develop a hypergraph WL kernel framework with two variants: the Hypergraph WL Subtree Kernel and Hypergraph WL Hyperedge Kernel. The Hypergraph WL Subtree Kernel counts different types of rooted subtrees and generates the final feature vector for a given hypergraph by comparing the number of different types of rooted subtrees. The Subtree Kernel identifies different rooted subtrees, while the Hyperedge Kernel focuses on hyperedges' vertex labels, enhancing feature vector generation. In order to fulfill our research objectives, a comprehensive set of experiments was meticulously designed, including seven graph classification datasets and 12 hypergraph classification datasets. Results on graph classification datasets indicate that the Hypergraph WL Subtree Kernel can achieve the same performance compared with the classical Graph Weisfeiler-Lehman Subtree Kernel. Results on hypergraph classification datasets show significant improvements compared to other typical kernel-based methods, which demonstrates the effectiveness of the proposed methods. In our evaluation, our proposed methods outperform the second-best method in terms of runtime, running over 80 times faster when handling complex hypergraph structures. This significant speed advantage highlights the great potential of our methods in real-world applications.