Model fingerprinting has emerged as a crucial mechanism for safeguarding the intellectual property of open-source models, offering a non-intrusive approach that requires no modifications to the protected model. However, our analysis reveals that existing fingerprinting techniques are fundamentally vulnerable to false claim attacks, wherein adversaries can fraudulently assert ownership over independent third-party models. We demonstrate that this vulnerability stems from the untargeted nature of current methods, which evaluate model similarity based on arbitrary sample outputs rather than alignment with a specific, predefined reference. To mitigate this vulnerability, we introduce FIT-Print, a targeted fingerprinting paradigm that actively counters false claim attacks. Specifically, FIT-Print leverages optimization to transform the fingerprint into a verifiable, targeted signature. Building upon this foundation, we propose two black-box fingerprinting methods, the bit-wise FIT-ModelDiff and the list-wise FIT-LIME, which utilize output distances and feature attributions as robust model signatures, respectively. Extensive evaluations across benchmark models and datasets show that our framework perfectly neutralizes false claim attacks (100% defense success rate) and eliminates false alarms on independent models (0.0%), all while maintaining a 100% ownership verification rate against diverse model reuse techniques.