Visual Place Recognition (VPR) is critical for autonomous navigation, yet state-of-the-art methods lack well-calibrated uncertainty estimation. Standard pipelines cannot reliably signal when a query is ambiguous or a match is likely incorrect, posing risks in safety-critical robotics. We propose KappaPlace, a principled framework for learning uncertainty-aware VPR representations. Our core contribution is a Prototype-Anchored supervision strategy that leverages latent class representatives as targets for a probabilistic objective. By modeling image descriptors as von Mises-Fisher (vMF) variables, we learn a lightweight module to predict the concentration parameter as a direct proxy for aleatoric uncertainty. While existing VPR uncertainty methods are typically restricted to a query-centric view, we derive a novel match-level formulation to quantify the reliability of specific query-reference pairs. Across five diverse benchmarks, KappaPlace reduces Expected Calibration Error (ECE@K) by up to 50% compared to existing methods while maintaining or improving retrieval recall. We provide both a joint-training variant and a post-training extension for frozen backbones. Our results demonstrate that KappaPlace provides a robust, stable, and well-calibrated signal that enables reliable decision-making within the VPR pipeline. Our code is available at: https://github.com/mayayank95/UncertaintyAwareVPR