A search for leptoquark pair production decaying into $te^-\bar{t}e^+$ or $t\mu^-\bar{t}\mu^+$ in final states with multiple leptons is presented. The search is based on a dataset of $pp$ collisions at $\sqrt{s}$=13 TeV recorded with the ATLAS detector during Run 2 of the Large Hadron Collider, corresponding to an integrated luminosity of 139 fb$^{-1}$. Events are selected with two or more light leptons (electron or muon) and at least two jets out of which at least one jet is identified as coming from a $b$-hadron. Four signal regions, with the requirement of at least three light leptons, are considered based on the number of leptons of a given flavour. The main background processes are estimated using dedicated control regions in a simultaneous fit with the signal regions to data. No excess above the Standard Model background prediction is observed and 95% confidence level limits on the production cross section times branching ratio are derived as a function of the leptoquark mass. Under the assumption of exclusive decays into $te^-$($t\mu^-$), the corresponding lower limit on the scalar mixed-generation leptoquark mass $m_{LQ_{\mathrm{mix}}^{\mathrm{d}}}$ is at 1.58 (1.59) TeV and on the vector leptoquark mass $m_{\tilde{U}_1}$ at 1.67 (1.67) TeV in the minimal coupling scenario and at 1.95 (1.95) TeV in the Yang-Mills scenario.