Backscatter communication (BC) emerges as a pivotal technology for ultra-low-power energy harvesting applications, but its practical deployment is often hampered by notable security vulnerabilities. Physical layer authentication (PLA) offers a promising solution for securing BC by leveraging the unique characteristics of the communication medium. However, existing PLA approaches often fall short due to limited signal strength in practical BC scenarios and performance deterioration with increasing distance between the tag and the reader. Moreover, achieving mutual authentication has been largely neglected in current PLA schemes, given the passive nature of tags and their limited computational and energy resources. This paper introduces a reconfigurable intelligent surfaces (RIS)-aided PLA scheme based on the physical features of received signals at legitimate endpoints through cascade links in monostatic BC (MBC) systems. By considering a RIS operating in its near-optimal conditions between a tag and a reader, the proposed PLA leverages the RIS-enhanced power delivery detected by the tag's energy detector and the optimized received signal strength (RSS) at the reader's signal processing unit, addressing the conventional challenges of mutual authentication, low PLA performance, and limited secure coverage area inherent in BC systems. Through theoretical analysis and extensive simulations, we show that as long as RIS is controlled by a trusted party in the network, it can boost the authentication performance across different system settings and strengthen the security features. Additionally, we analyze to explore the potential security threats when the RIS is compromised by an adversary, assessing its impact on the system's PLA performance and secrecy capacity, providing a comprehensive understanding of the security implications for RIS-aided MBC under such circumstances.