This Letter presents the observation of the electroweak production of two isolated photons in association with two jets together with measurements of the corresponding fiducial and differential cross sections. The analysis is performed using proton-proton collision data collected by ATLAS at the LHC at $\sqrt{s}$ = 13 TeV and with an integrated luminosity of 140 fb$^{-1}$. The electroweak production of two isolated photons in association with two jets is observed with a significance of $6.2 σ$ and with a fiducial cross-section of $σ^{\mathrm{EW}}_{\mathrm{obs}}= 13.8^{+3.0}_{-2.6}\, \mathrm{fb}$, in agreement with the Standard Model prediction of $σ^{\mathrm{EW}}_{\mathrm{obs}}= 17.1^{+2.4}_{-2.4}\, \mathrm{fb}$. Differential cross-sections are measured as a function of various kinematical variables probing photon and jet properties and compared with predictions. In addition, fiducial and differential cross-sections are reported for the combined QCD and electroweak production of two isolated photons and two jets. The measurement is also interpreted in the context of an effective field theory, and limits are set on the Wilson coefficients of several dimension-8 operators parameterising anomalous quartic gauge couplings.