Self-hybridized strong coupling between optical modes and excitons within the same material has garnered significant attention because of its potential impact on the development of new technologies using simple, single-material approaches and without the need for an external cavity. While self-hybridization can be achieved in various kinds of natural materials, we explore the possibility of realizing the phenomenon in nanoparticles composed of artificial, hyperbolic metamaterials. Using a base of silver-silica multilayer as the base nanoresonator, we characterize its quasinormal mode (QNM) structure and subsequently dope the dielectric layers with excitonic transitions described by a Lorentzian permittivity. We explore the propensity of uniaxial hyperbolic nanoresonators to achieve self-hybridization with excitons built into their material matrix as a function of their properties. The results show that hyperbolic nanoparticles are capable of supporting strong coupling for both the electric and magnetic dipole modes even at low oscillator strengths due to a strongly suppressed QNM volume by up to two orders of magnitude over their geometrical one.