User identity linkage (UIL), matching accounts of a person on different social networks, is a fundamental task in cross-network data mining. Recent works have achieved promising results by exploiting graph neural networks (GNNs) to capture network structure. However, they rarely analyze the realistic node-level bottlenecks that hinder UIL's performance. First, node degrees in a graph vary widely and are long-tailed. A significant fraction of tail nodes with small degrees are underrepresented due to limited structural information, degrading linkage performance seriously. The second bottleneck usually overlooked is super head nodes. It is commonly accepted that head nodes perform well. However, we find that some of them with super high degrees also have difficulty aligning counterparts, due to noise introduced by the randomness of following friends in real-world social graphs. In pursuit of learning ideal representations for these two groups of nodes, this paper proposes a degree-aware model named DegUIL to narrow the degree gap. To this end, our model complements missing neighborhoods for tail nodes and discards redundant structural information for super head nodes in embeddings respectively. Specifically, the neighboring bias is predicted and corrected locally by two modules, which are trained using the knowledge from structurally adequate head nodes. As a result, ideal neighborhoods are obtained for meaningful aggregation in GNNs. Extensive experiments demonstrate the superiority of our model. Our data and code can be found at https://github.com/Longmeix/DegUIL.