Subgraph matching query is a fundamental problem in graph data management and has a variety of real-world applications. Several recent works utilize deep learning (DL) techniques to process subgraph matching queries. Most of them find approximate subgraph matching results without accuracy guarantees. Unlike these DL-based inexact subgraph matching methods, we propose a learning-based exact subgraph matching framework, called \textit{graph neural network (GNN)-based anchor embedding framework} (GNN-AE). In contrast to traditional exact subgraph matching methods that rely on creating auxiliary summary structures online for each specific query, our method indexes small feature subgraphs in the data graph offline and uses GNNs to perform graph isomorphism tests for these indexed feature subgraphs to efficiently obtain high-quality candidates. To make a tradeoff between query efficiency and index storage cost, we use two types of feature subgraphs, namely anchored subgraphs and anchored paths. Based on the proposed techniques, we transform the exact subgraph matching problem into a search problem in the embedding space. Furthermore, to efficiently retrieve all matches, we develop a parallel matching growth algorithm and design a cost-based DFS query planning method to further improve the matching growth algorithm. Extensive experiments on 6 real-world and 3 synthetic datasets indicate that GNN-AE is more efficient than the baselines, especially outperforming the exploration-based baseline methods by up to 1--2 orders of magnitude.