A search is presented for high-mass exclusive diphoton production via photon-photon fusion in proton-proton collisions at$\sqrt{s}=13\mathrm{TeV}$in events where both protons survive the interaction. The analysis utilizes data corresponding to an integrated luminosity of$103{\mathrm{fb}}^{-1}$collected in 2016–2018 with the central CMS detector and the CMS and TOTEM precision proton spectrometer (PPS). Events that have two photons with high transverse momenta ($p_{\mathrm{T}}^{\gamma }>100\mathrm{GeV}$), back-to-back in azimuth, and with a large diphoton invariant mass ($m_{\gamma \gamma }>350\mathrm{GeV}$) are selected. To remove the dominant inclusive diphoton backgrounds, the kinematic properties of the protons detected in PPS are required to match those of the central diphoton system. Only events having opposite-side forward protons detected with a fractional momentum loss between 0.035 and 0.15 (0.18) for the detectors on the negative (positive) side of CMS are considered. One exclusive diphoton candidate is observed for an expected background of 1.1 events. Limits at 95% confidence level are derived for the four-photon anomalous coupling parameters$|{\zeta }_1|<0.073{\mathrm{TeV}}^{-4}$and$|{\zeta }_2|<0.15{\mathrm{TeV}}^{-4}$, using an effective field theory. Additionally, upper limits are placed on the production of axionlike particles with coupling strength to photons$f^{-1}$that varies from$0.03{\mathrm{TeV}}^{-1}$to$1{\mathrm{TeV}}^{-1}$over the mass range from 500 to 2000 GeV.