Properties of the Higgs boson are measured in the two-photon final state using 36.1 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ of proton-proton collision data recorded at $s=13$ TeV by the ATLAS experiment at the Large Hadron Collider. Cross-section measurements for the production of a Higgs boson through gluon-gluon fusion, vector-boson fusion, and in association with a vector bosonor a top-quark pair are reported. The signal strength, defined as the ratio of the observed to the expected signal yield, is measured for each of these production processes as well as inclusively. The global signal strength measurement of $0.99\pm 0.14$ improves on the precision of the ATLAS measurement at $s=7$ and 8 TeV by a factor of two. Measurements of gluon-gluon fusion and vector-boson fusion productions yield signal strengths compatible with the Standard Model prediction. Measurements of simplified template cross sections, designed to quantify the different Higgs boson production processes in specific regions of phase space, are reported. The cross section for the production of the Higgs boson decaying to two isolated photons in a fiducial region closely matching the experimental selection of the photons is measured to be $55\pm 10$ fb, which is in good agreement with the Standard Model prediction of $64\pm 2$ fb. Furthermore, cross sections in fiducial regions enriched in Higgs boson production in vector-boson fusion or in association with large missing transverse momentum, leptons or top-quark pairs are reported. Differential and double-differential measurements are performed for several variables related to the diphoton kinematics as well as the kinematics and multiplicity of the jets produced in association with a Higgs boson. No significant deviations from a wide array of Standard Model predictions are observed.