  Continuous symmetries are believed to emerge at many quantum critical points
in frustrated magnets. In this work, we study two candidates of this paradigm:
the transverse-field frustrated Ising model (TFFIM) on the triangle and the
honeycomb lattices. The former is the prototypical example of this paradigm,
and the latter has recently been proposed as another realization. Our
large-scale Monte Carlo simulation confirms that the quantum phase transition
(QPT) in the triangle lattice TFFIM indeed hosts an emergent O(2) symmetry, but
that in the honeycomb lattice TFFIM is a first-order QPT and does not have an
emergent continuous symmetry. Furthermore, our analysis of the order parameter
histogram reveals that such different behavior originates from the irrelevance
and relevance of anisotropic terms near the QPT in the low-energy effective
theory of the two models. The comparison between theoretical analysis and
numerical simulation in this work paves the way for scrutinizing investigation
of emergent continuous symmetry at classical and quantum phase transitions.
