Service robots must infer object ownership to correctly interpret instructions such as "bring me my cup." However, ownership is a latent attribute that cannot be directly observed, and existing methods often rely on limited cues such as recent usage, making them unreliable in scenarios such as temporary sharing. We propose a framework for context-aware ownership inference with uncertainty-guided interaction (COIN). The method integrates user background information and object usage history using a large language model (LLM) to estimate ownership scores. To handle uncertainty, we apply conformal prediction to construct a set of plausible owners and selectively generate user queries when the prediction is uncertain. Experiments in a simulated home environment show that the proposed method consistently outperforms baseline approaches, achieving a Subset Accuracy of 0.988 and a Mean Jaccard index of 0.991. The method also maintains high performance in scenarios involving temporary use and shared ownership. The results demonstrate that combining contextual reasoning with uncertainty-aware interaction improves both estimation accuracy and robustness. The project page is available at https://emergentsystemlabstudent.github.io/COIN/.