The ratio of the cross sections for inclusive isolated-photon production in $pp$ collisions at centre-of-mass energies of 13 and 8 TeV is measured using the ATLAS detector at the LHC. The integrated luminosities of the 13 TeV and 8 TeV datasets are 3.2 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ and 20.2 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$, respectively. The ratio is measured as a function of the photon transverse energy in different regions of the photon pseudorapidity. The predictions from next-to-leading-order perturbative QCD calculations are compared with the measured ratio. The experimental systematic uncertainties as well as the uncertainties affecting the predictions are evaluated taking into account the correlations between the two centre-of-mass energies, resulting in a reduction of up to a factor of $2.5$ ($5$) in the experimental (theoretical) systematic uncertainties. The predictions based on several parameterisations of the proton parton distribution functions agree with the data within the reduced experimental and theoretical uncertainties. In addition, this ratio to that of the fiducial cross sections for $Z$ boson production at 13 and 8 TeV using the decay channels $Z\to e+e-$ and $Z\to \mu +\mu -$ is made and compared with the theoretical predictions. In this double ratio, a further reduction of the experimental uncertainty is obtained because the uncertainties arising from the luminosity measurement cancel out. The predictions describe the measurements of the double ratio within the theoretical and experimental uncertainties.