Hyperbolic nanoresonators, composed of anisotropic materials with opposite signs of permittivity, have unique optical properties due to a large degree of freedom that hyperbolic dispersion provides in designing their response. Here, we focus on uniaxial hyperbolic nanoresonators composed of a model silver-silica multilayer in the form of spheroids with a broad aspect ratio encompassing both prolate and oblate particles. The origin and evolution of the optical response and mode coupling are investigated using both numerical (T-matrix and FDTD) and theoretical methods. We show the tunability of the optical resonances and the interplay of the shape and material anisotropy in determining the spectral response. Depending on the illumination conditions as well as shape and material anisotropy, a single hyperbolic spheroid can show a dominant electric resonance, behaving as a pure metallic nanoparticle, or a strong dipolar magnetic resonance even in the quasistatic regime. The quasistatic magnetic response of indicates a material-dependent origin of the mode, which is obtained due to coupling of the magnetic and electric multipoles. Such coupling characteristics can be employed in various modern applications based on metasurfaces.