We investigate single- and two-parameter metrological responses near second-order critical boundaries and triple points in the anisotropic quantum Rabi model. In the large-frequency-ratio limit, approaching a second-order critical boundary leads to divergent ground-state squeezing and path-coordinate quantum Fisher information (QFI), whereas both remain finite along fixed-direction approaches to a triple point. However,the Gaussian ground state depends on only one effective squeezing parameter and therefore cannot distinguish two independent coupling variations. To achieve two-parameter identifiability, we employ a coherent initial state in the parameter encoding. Near a second-order critical boundary, the residual quadratic dynamics enhances sensitivity but generally also introduces measurement incompatibility. In the triple-point limit, dynamical quadrature mixing is suppressed, while the coherent displacement preserves two independent parameter responses. Thus, the two coupling parameters remain locally identifiable while the mean symmetric logarithmic derivative (SLD) commutator tends to zero. We further derive an exact compatibility locus inside the normal phase. Along this locus, the two coupling parameters remain locally identifiable, and the Holevo and SLD Cram'er–Rao bounds coincide. Full-model calculations at finite frequency ratio support these finite-time results. These results clarify the distinct roles of critical sensitivity, parameter identifiability, and measurement compatibility in multiparameter sensing.