<p>We define anisotropic surface energy using a non-negative gauge function, instead of the conventional Minkowski norm, to study anisotropic capillary hypersurfaces within a wedge in Euclidean space. This novel framework helps to illuminate a profound connection between capillary hypersurfaces and free boundary hypersurfaces. Our main results include new Minkowski formulae and a Heintze-Karcher type inequality. Furthermore, we apply these results to establish an Alexandrov-type theorem, thereby extending known isotropic results to the anisotropic setting. A key aspect of our proof involves a subtle analysis along the intersection of a hypersurface and the boundary of the wedge.</p>