The domain-area distribution in the phase transition dynamics of Z₂ symmetry breaking is studied theoretically and numerically for phase-separating two-component Bose-Einstein condensates in quasi-two dimensions. According to the dynamic scaling law of the phase ordering kinetics, the domain-area distribution is described by a universal function of the domain area, rescaled by the mean inter-domain-wall distance. The scaling theory for general coarsening dynamics hypothesizes that the distribution during the coarsening dynamics has a hierarchy with the two scaling regimes, the microscopic and macroscopic regimes with distinct power-law exponents. The power law in the latter regime, where the domain size is larger than the mean distance, is universally represented with the Fisher's exponent of the percolation theory in two dimensions. On the other hand, the power-law exponent in the microscopic regime is sensitive to the microscopic dynamics of the system. This conjecture is confirmed by large-scale numerical simulations of the coupled Gross-Pitaevskii equation for binary condensates. In the numerical experiments of the superfluid system, the exponent in the microscopic regime anomalously reaches to its theoretical upper limit of the general scaling theory. The anomaly comes from the quantum-fluid effect in the presence of circular vortex sheets, described by the hydrodynamic approximation neglecting the fluid compressibility. It is also found that the distribution of superfluid circulation along vortex sheets obeys a dynamic scaling law with different power-law exponents in the two regimes.<br>[1] Hiromitsu Takeuchi, Phys. Rev. A 97, 013617 (2018).<br>[2] Hiromitsu Takeuchi, J. Low Temp. Phys. to be published (2019).