Complex networks provide a crucial framework for understanding the structure and dynamic behavior of real-world systems, while consensus and controllability serve as the foundational theoretical cornerstones for achieving coordinated control and global behavior regulation in networks. In this paper, the consensus and controllability of multi-agent systems on a double-layer star topology are studied from an algebraic perspective. This study systematically investigates the consensus problem of undirected double-layer star networks under the constraint that intra-layer coupling strengths are equal based on a continuous first-order linear protocol framework, and establishes the controllability conditions for this double-layer network system by using the PBH test based on the identical intra-layer and inter-layer coupling strengths. To validate the correctness and effectiveness of the theoretical results, numerical simulations are conducted by constructing several test cases for verification. This work establishes a theoretical analysis framework for multi-agent systems on double-layer star networks, which can provide deeper insights into coordination control of complex systems.