The photon-induced production of a pair of tau leptons is measured in proton-proton collisions. The analysis uses a data set recorded with the CMS detector at the LHC at a center-of-mass energy of 13 TeV and corresponding to an integrated luminosity of 138 fb−1. Events with a small number of tracks close to the di-tau vertex are selected to isolate the exclusive production mechanism. Limits are set on the anomalous magnetic moment of the tau lepton, improving by a factor of 5 the constraints set during the LEP era.