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An attributed hypergraph comprises nodes with attributes and hyperedges that connect varying numbers of nodes. Attributed hypergraph node and hyperedge embedding (AHNEE) maps nodes and hyperedges to compact vectors for use in important tasks such as node classification, hyperedge link prediction, and hyperedge classification. Generating high-quality embeddings is challenging due to the complexity of attributed hypergraphs and the need to embed both nodes and hyperedges, especially in large-scale data. Existing solutions often fall short by focusing only on nodes or lacking native support for attributed hypergraphs, leading to inferior quality, and struggle with scalability on large attributed hypergraphs. We propose SAHE, an efficient and effective approach that unifies node and hyperedge embeddings for AHNEE computation, advancing the state of the art via comprehensive embedding formulations and algorithmic designs. First, we introduce two higher-order similarity measures, HMS-N and HMS-E, to capture similarities between node pairs and hyperedge pairs, respectively. These measures consider multi-hop connections and global topology within an extended hypergraph that incorporates attribute-based hyperedges. SAHE formulates the AHNEE objective to jointly preserve all-pair HMS-N and HMS-N similarities. Direct optimization is computationally expensive, so we analyze and unify core approximations of all-pair HMS-N and HMS-N to solve them simultaneously. To enhance efficiency, we design several non-trivial optimizations that avoid iteratively materializing large dense matrices while maintaining high-quality results. Extensive experiments on diverse attributed hypergraphs and 3 downstream tasks, compared against 11 baselines, show that SAHE consistently outperforms existing methods in embedding quality and is up to orders of magnitude faster.