High Harmonic Generation (HHG) in solids often carries the microstructure information of crystal materials, and is a practical means to detect the electronic properties and dynamics in crystals. The anomalous dependence of solid HHG on the ellipticity of driving laser has received much attention. As is well known, gas HHG is produced by the recollision of electron with parent nucleus, the recollision probability decreases greatly with the increase of ellipticity, so the harmonic intensity driven by linearly polarized light is always larger than that driven by elliptically polarized light. A large number of experiments and theories have shown that the anomalous ellipticity dependence appears in the HHG from graphene and surface states of topological insulators with Dirac-cone structure, which is different from the gas HHG. YOSHIKAWA N et al. think that the ellipticity dependence of HHG in graphene is determined by the comparison of bandgap and Rabi frequency. However, in Bi₂Se₃, a topological insulator with a similar Dirac cone on two-dimensional surface state, BAYKUSHEVA D et al. believe that the mechanism of its anomalous ellipticity dependence is different from that of graphene. Topological insulators require strong Spin-Orbit Coupling (SOC) as a prerequisite, however, the SOC in graphene is weak. They conclude that the anomalous ellipticity associated with band topology is dependent on the strong SOC and protection of time-reversal symmetry. Therefore, both Dirac cone and SOC effect have influence on the harmonic ellipticity dependence, and there may be competition between them, thus the mechanism of the anomalous ellipticity dependence of Dirac system remains unclear.In this paper, the HHG of the graphene and surface states of topological insulators under different laser ellipticities is simulated by solving semiconductor Bloch equations. The anomalous ellipticity dependence of HHG from graphene and surface state of topological insulator is theoretically verified. We investigate the microscopic dynamics of the electron transition near the Dirac cone under different laser ellipticities and the transient changes of the harmonic radiation in k space. The gapless and linear-dispersion Dirac cone in linear polarization will form a forbidden transition path in k space, which is the micro-mechanism of the lower HHG yield of Dirac system driven by linearly polarized laser than the elliptical case. By adding SOC to a tight-binding model of graphene to open the bandgap around its Dirac cone, we find that the enhancement of SOC does not induce the anomalous ellipticity dependence of HHG. In addition, the laser intensity can change the profile of harmonic ellipticity dependence, but it cannot determinate the appearance of anomalous ellipticity dependence, which can be attributed to the gapless and linear-dispersion Dirac cone in essence. This work clarifies the physical origin of anomalous ellipticity dependence in the Dirac system and establishes a direct link between the Dirac cone and the anomalous ellipticity dependence of HHG in solids, which is conducive to promoting optical characterization of Dirac materials and the detection of ultrafast nonlinear optical properties of Dirac electrons by the HHG. The abnormal ellipticity dependence of Dirac material breaks through the monotonic decrease of harmonic intensity in gas with the increase of laser ellipticity, which is expected to be applied to the development of new optical sensors or optoelectronic devices.