Multivariate conformal prediction requires nonconformity scores that compress residual vectors into scalars while preserving certain implicit geometric structure of the residual distribution. We introduce a Multivariate Kernel Score (MKS) that produces prediction regions that explicitly adapt to this geometry. We show that the proposed score resembles the Gaussian process posterior variance, unifying Bayesian uncertainty quantification with the coverage guarantees of frequentist-type. Moreover, the MKS can be decomposed into an anisotropic Maximum Mean Discrepancy (MMD) that interpolates between kernel density estimation and covariance-weighted distance. We prove finite-sample coverage guarantees and establish convergence rates that depend on the effective rank of the kernel-based covariance operator rather than the ambient dimension, enabling dimension-free adaptation. On regression tasks, the MKS reduces the volume of prediction region significantly, compared to ellipsoidal baselines while maintaining nominal coverage, with larger gains at higher dimensions and tighter coverage levels.