We experimentally demonstrate the strong coupling between the ferromagnetic magnons in an yttrium-iron-garnet (YIG) sphere and the drive-field-induced dressed states of a superconducting qubit, which gives rise to the double dressing of the superconducting qubit. The YIG sphere and the superconducting qubit are embedded in a microwave cavity, and are coupled to the magnetic and electrical fields of the cavity $\mathrm{TE}_{102}$ mode, respectively. The effective coupling between them is mediated by the virtual cavity photons of cavity $\mathrm{TE}_{102}$ mode. Our experimental results indicate that as the power for driving the qubit increases, an additional split of the qubit-magnon polariton occurs. These supplemental splittings indicate a double-dressed state. We theoretically analyze the experimental results by using a particle-hole symmetric model. The theoretical results fit the experimental observations well in a broad range of drive-field power parameters, revealing that the driven qubit-magnon hybrid quantum system can be used to emulate a particle-hole symmetric pair coupled to a bosonic mode. Our hybrid quantum system holds great promise for quantum simulations of composite quasiparticles consisting of fermions and bosons.