The efficiency of the photon identification criteria in the ATLAS detector is measured using 36.1 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ to 36.7 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ of $pp$ collision data at $s$ = 13 TeV collected in 2015 and 2016. The efficiencies are measured separately for converted and unconverted isolated photons, in four different pseudorapidity regions, for transverse momenta between 10 GeV and 1.5 TeV. The results from the combination of three data-driven techniques are compared with the predictions from simulation after correcting the variables describing the shape of electromagnetic showers in simulation for the average differences observed relative to data. Data-to-simulation efficiency ratios are determined to account for the small residual efficiency differences. These factors are measured with uncertainties between 0.5% and 5% depending on the photon transverse momentum and pseudorapidity. The impact of the isolation criteria on the photon identification efficiency, and that of additional soft $pp$ interactions, are also discussed. The probability of reconstructing an electron as a photon candidate is measured in data, and compared with the predictions from simulation. The efficiency of the reconstruction of photon conversions is measured using a sample of photon candidates from $Z\to \mu \mu \gamma$ events, exploiting the properties of the ratio of the energies deposited in the first and second longitudinal layers of the ATLAS electromagnetic calorimeter.