Entity alignment (EA) aims to match identical entities across different knowledge graphs (KGs). Graph neural network-based entity alignment methods have achieved promising results in Euclidean space. However, KGs often contain complex local and hierarchical structures, which are hard to represent in a single space. In this paper, we propose a novel method named as UniEA, which unifies dual-space embedding to preserve the intrinsic structure of KGs. Specifically, we simultaneously learn graph structure embeddings in both Euclidean and hyperbolic spaces to maximize the consistency between embeddings in the two spaces. Moreover, we employ contrastive learning to mitigate the misalignment issues caused by similar entities, where embeddings of similar neighboring entities become too close. Extensive experiments on benchmark datasets demonstrate that our method achieves state-of-the-art performance in structure-based EA. Our code is available at https://github.com/wonderCS1213/UniEA.