A Physical Unclonable Function (PUF) is a promising concept for device authentication, as it can be extracted from existing device components and enables the generation of unique, device-specific keys. However, existing PUF-based approaches often require prior distribution of these keys during an enrolment phase, require persistent storage or transmission of sensitive data, and are vulnerable to different attacks. To address these limitations, we propose a novel approach to using PUFs. While PUFs are typically employed by leveraging the inherent uniqueness of their responses, our approach instead focuses on their similarities, exploiting matching segments of two distinct PUFs. This enables the identification of a PUF that is unique per-device-pair, termed Inter-Device PUF (ID-PUF), which links two devices and serves as the basis for a shared secret key. This key can be independently generated on both devices, eliminating the need for its prior distribution and persistent storage, thus making an ID-PUF well-suited for application in lightweight symmetric cryptographic schemes. The feasibility of an ID-PUF is demonstrated through statistical analysis and experimental evaluation on several devices. Furthermore, to illustrate its application, we present a novel ID-PUF-based mutual authentication protocol (ID-PUMA) and analyse its security.