We present a search for magnetic monopoles and high-electric-charge objects using LHC Run 2 $ \sqrt{s} $ = 13 TeV proton-proton collisions recorded by the ATLAS detector. A total integrated luminosity of 138 fb$^{−1}$ was collected by a specialized trigger. No highly ionizing particle candidate was observed. Considering the Drell-Yan and photon-fusion pair production mechanisms as benchmark models, cross-section upper limits are presented for spin-0 and spin-1/2 magnetic monopoles of magnetic charge 1g$_{D}$ and 2g$_{D}$ and for high-electric-charge objects of electric charge 20 ≤ |z| ≤ 100, for masses between 200 GeV and 4000 GeV. The search improves by approximately a factor of three the previous cross-section limits on the Drell-Yan production of magnetic monopoles and high-electric charge objects. Also, the first ATLAS limits on the photon-fusion pair production mechanism of magnetic monopoles and high-electric-charge objects are obtained.