The Dicke model (DM) serves as a paradigm for understanding collective light-matter interactions. We introduce the chiral Dicke model, a generalization where an atomic ensemble couples to a two-mode cavity via chiral interactions. Unlike the standard DM, the chiral DM is endowed with an inherent continuous $U(1)$ symmetry associated with angular momentum conservation. The ground-state phase diagram and the associated quantum phase transitions are charted out, revealing a $U(1)$-broken superradiant phase that spans a broad parameter space. We demonstrate that the spectrum of quantum fluctuations is tunable in both the symmetric and broken phases. Notably, the system exhibits multiversality, a form of path-dependent criticality in which distinct universality classes govern the transition between the same two phases, depending on where the critical surface is crossed. Specifically, the critical exponents of the normal-superradiant transition take the values of the standard DM on a codimension-one subset of the critical surface, and those of the $SO(2)$ Lipkin-Meshkov-Glick (LMG) model elsewhere. This multiversality has a sharp dynamical signature: a slow ramp from the normal phase to the critical surface generates heat with universal Kibble-Zurek scaling at the Dicke-type critical points, but only analytic scaling with the ramp rate at the LMG-type ones. Our work establishes the chiral Dicke model as a powerful platform to realize novel quantum phases and multiversality in light-matter coupled systems.