We present the high-precision magnetization data of the valence fluctuating heavy fermion superconductor $beta$-YbAlB$_4$ in a wide temperature range from 0.02 K to 320 K spanning four orders of magnitude. We made detailed analyses of the $T/B$ scaling of the magnetization, and firmly confirmed the unconventional zero-field quantum criticality (QC) without tuning. We examined other possible scaling relationship such as $T/(B-B_c)^{delta}$ scaling, and confirmed that $delta = 1$ provides the best quality of the fit with an upper bound on the critical magnetic field $vert B_c vert <0.2$~mT. We further discuss the heavy Fermi-liquid component of the magnetization after subtracting the QC component estimated based on the $T/B$ scaling. The temperature dependence of the heavy Fermi-liquid component is found very similar to the magnetization of the polymorph $alpha$-YbAlB$_4$. In addition, the heavy Fermi-liquid component is suppressed in the magnetic field above $sim$ 5 T as in $alpha$-YbAlB$_4$. This was also confirmed by the magnetization measurements up to $sim 50$ T for both $alpha$- and $beta$-YbAlB$_4$. Interestingly, the detailed analyses revealed that the only a part of $f$ electrons participates in the zero-field QC and the heavy fermion behavior. We also present a temperature - magnetic field phase diagram of ybal to illustrate how the characteristic temperature and field scales evolves near the QC.
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