  We study the quantum multicritical point in a (2+1)-dimensional Dirac system
between the semimetallic phase and two ordered phases that are characterized by
anticommuting mass terms with $O(N_1)$ and $O(N_2)$ symmetry, respectively.
Using $\epsilon$ expansion around the upper critical space-time dimension of
four, we demonstrate the existence of a stable renormalization-group fixed
point, enabling a direct and continuous transition between the two ordered
phases directly at the multicritical point. This point is found to be
characterized by an emergent $O(N_1+N_2)$ symmetry for arbitrary values of
$N_1$ and $N_2$ and fermion flavor numbers $N_f$, as long as the corresponding
representation of the Clifford algebra exists. Small $O(N)$-breaking
perturbations near the chiral $O(N)$ fixed point are therefore irrelevant. This
result can be traced back to the presence of gapless Dirac degrees of freedom
at criticality, and it is in clear contrast to the purely bosonic $O(N)$ fixed
point, which is stable only when $N < 3$. As a by-product, we obtain
predictions for the critical behavior of the chiral $O(N)$ universality classes
for arbitrary $N$ and fermion flavor number $N_f$. Implications for critical
Weyl and Dirac systems in 3+1 dimensions are also briefly discussed.
