The fundamental semiconductor concept of doping has recently been transplanted to photonics in the platform of epsilon-near-zero (ENZ) media. By doping nonmagnetic impurities, ENZ media can exhibit almost arbitrary magnetism. However, this original photonic doping approach results only in isotropic media and thus cannot achieve impedance matching for all incident angles. We extend the photonic doping approach of ENZ media by adding anisotropy, which enables full transparency with omnidirectional impedance matching. More importantly, such anisotropically doped ENZ media preserve their material parameters under arbitrary coordinate transformations, thereby providing a powerful platform to construct various ideal transformation optical devices. As an example, a full-parameter omnidirectional invisibility cloak is demonstrated to hide objects from a wide range of incident angles. The transformation-invariant material proposed not only supplements and extends the rising technologies of ENZ media but also constitutes a significant step towards the practical implementation of ideal transformation optical devices.