  We present results for the nucleon form factors: electric ($G_E$), magnetic
($G_M$), axial ($F_A$), induced pseudoscalar ($F_P$) and pseudoscalar ($G_P$)
form factors, using the second PACS10 ensemble that is one of three sets of
$2+1$ flavor lattice QCD configurations at physical quark masses in large
spatial volumes (exceeding $(10\ \mathrm{fm})^3$). The second PACS10 gauge
configurations are generated by the PACS Collaboration with the six
stout-smeared $O(a)$ improved Wilson quark action and Iwasaki gauge action at
the second gauge coupling $\beta=2.00$ corresponding to the lattice spacing of
$a=0.063$ fm. We determine the isovector electric, magnetic and axial radii and
magnetic moment from the corresponding form factors, as well as the
axial-vector coupling $g_A$. Combining our previous results for the coarser
lattice spacing [E. Shintani et al., Phys. Rev. D99 (2019) 014510; Phys. Rev.
D102 (2020) 019902 (erattum)], the finite lattice spacing effects on the
isovector radii, magnetic moment and axial-vector coupling are investigated
using the difference between the two results. It was found that the effect on
$g_A$ is kept smaller than the statistical error of 2% while the effect on the
isovector radii was observed as a possible discretization error of about 10%,
regardless of the channel. We also report the partially conserved axial vector
current (PCAC) relation using a set of nucleon three-point correlation
functions in order to verify the effect by $O(a)$-improvement of the
axial-vector current.
