Multilayer network node classification is a critical task in complex network research. It aims to identify and classify nodes within a network to uncover their intrinsic multidimensional structure and semantic characteristics. Graph Neural Networks (GNNs) , as a framework capable of directly process graph-structured data, can effectively capture the underlying patterns of nodes and edges, as well as more profound semantic features, through message-passing mechanisms. In node classification tasks, GNNs exploit graph structure information and node features by aggregating information from neighboring nodes and propagating messages. Compared to traditional node classification methods, GNNs learn node feature representations adaptively, reducing reliance on manual feature engineering and enhancing classification accuracy. As a result, GNN-based approaches have become the mainstream in node classification research. This study categorized and reviewed recent developments in GNN-based multilayer network node classification methods. First, it outlined the core concepts of multilayer networks and GNNs, and provided definitions and characteristics of various multilayer network models. Then, it comprehensively summarized the latest advancements in GNN-based multilayer network node classification methods, and categorized them into three paradigms based on the learning approach: semi-supervised, unsupervised, and self-supervised learning. The paper also analyzes the performance of these methods in downstream tasks, such as social networks, academic networks, and e-commerce platforms. Finally, it provides a thorough summary of existing research, discusses the limitations of current methods, and suggests potential directions for future research.