Channel charting enables location-based services (LBSs) without requiring explicit position information by using pseudo-locations from the channel chart. While this property implies inherent privacy advantages, it does not provide formal privacy guarantees. In this work, we address location privacy in channel charting referred to as chart location indistinguishability (CLI), which extends geo-indistinguishability (GI) to channel charting representations. In order to achieve CLI, a standard planar Laplace mechanism is investigated and a geometry-aware Mahalanobis norm planar Laplace (MNPL) mechanism is devised. The proposed MNPL mechanism perturbs the channel chart by injecting noise aligned with the local structure of the chart. In the CLI framework with MNPL, privacy is defined in latent channel chart manifolds using locally adaptive covariance derived from chart neighborhoods, while preserving manifold topology under privacy constraints. In addition, differential privacy is considered as a privacy baseline. The proposed approach is evaluated across multiple channel charting schemes. The performance is assessed using utility metrics such as quality loss (QL) and range query error (RQE), as well as geometry-aware metrics including trustworthiness (TW) and continuity (CT). Numerical results demonstrate that the proposed privacy mechanism provides strong privacy guarantees while preserving the channel chart for LBSs tasks.