In the picture of molecular cavity optomechanics, surface-enhanced Raman scattering (SERS) can be understood as molecular oscillators parametrically coupled to plasmonic nanocavities supporting an extremely localized optical field. This enables SERS from conventional fingerprint detection toward quantum nanotechnologies associated with, e.g., frequency upconversion and optomechanically induced transparency. Here, we study a phononic cavity optomechanical system consisting of a monolayer MoS2 placed inside a plasmonic nanogap, where the coherent phonon-plasmon interaction involves the collective oscillation from tens of thousands of unit cells of the MoS2 crystal. We observe the selective nonlinear SERS enhancement of the system as determined by the laser-plasmon detuning, suggesting the dynamic backaction modification of the phonon populations. Anomalous superlinear power dependence of a second-order Raman-inactive phonon mode with respect to the first-order phonons is also observed, indicating the distinctive properties of the phononic nanodevice compared with the molecular system. Our results promote the development of robust phononic optomechanical nanocavities to further explore the related quantum correlation and nonlinear effects including parametric instabilities.