In very recent times, graph neural network (GNN) and network/graph representation learning (NRL) (a.k.a. network/graph embedding) have been considered as the hottest research topics which have attracted attentions from researchers in multiple disciplines. Given the wide range of applications of network representation learning (NRL), numerous studies have been devoted to this field, demonstrating significant improvements in both the quality of network representations and the effectiveness of task-driven fine-tuning for various downstream tasks such as node clustering and classification. However, most recent attributed graph embedding methods still lack a comprehensive focus on integrating multi-view embeddings that combine both relational and attribute features within complex, structure-rich graphs. Furthermore, existing approaches often overlook the shared neighborhood correlations among nearby nodes during the embedding process. To address these limitations, this paper introduces SEAGE, which is a novel structure-enhanced attributed graph embedding method. Our proposed SEAGE model can assist to improve joint structural and attribute-based representation learning by incorporating a self-attention mechanism, which softly aligns the two feature types into a unified and task-adaptive embedding space. The resulting rich embeddings are then fed into a multi-layer GNN-based architecture to capture global structural representations. Finally, the context-aware node embeddings are processed through a fully connected layer to perform node classification in a supervised learning manner. Extensive experiments on benchmark graph-structured datasets validate the effectiveness of our proposed SEAGE model while handling different graph learning problems, such as node classification and link prediction.