The recently proposed Graph Matching Network models (GMNs) effectively improve the inference accuracy of graph similarity analysis tasks. GMNs often take graph pairs as input, embed nodes features, and match nodes between graphs for similarity analysis. While GMNs deliver high inference accuracy, the all-to-all node matching stage in GMNs introduces quadratic computing complexity with excessive memory accesses, resulting in significant computing and memory overhead that cannot be handled by existing approaches. In this paper, we propose the Coordinated Elastic Graph Matching Accelerator (CEGMA), a software and hardware co-design accelerator to address the challenges of GMNs. Specifically, by exploiting duplicate subgraphs in the input graphs, we develop an elastic matching filter to significantly reduce the quadratic computing overhead. By exploring the substantial data reuses oriented from accessing node features, we propose a cross-graph coordinator that fuses cross-graph similarity computing with intra-graph computing to enhance data locality. Experimental results show that, on average, CEGMA achieves 353× and 6.5× speedups in GMN computing compared to state-of-the-art GPU implementation and GNN accelerators, respectively.