Several observables sensitive to the fragmentation of $b$-quarks into $b$-hadrons are measured using 36 fb${}^{-1}$ of $\sqrt s = 13$ TeV proton-proton collision data collected with the ATLAS detector at the LHC. Jets containing $b$-hadrons are obtained from a sample of dileptonic $t\bar{t}$ events, and the associated set of charged-particle tracks is separated into those from the primary $pp$ interaction vertex and those from the displaced $b$-decay secondary vertex. This division is used to construct observables that characterize the longitudinal and transverse momentum distributions of the $b$-hadron within the jet. The measurements have been corrected for detector effects and provide a test of heavy-quark-fragmentation modeling at the LHC in a system where the top-quark decay products are color-connected to the proton beam remnants. The unfolded distributions are compared with the predictions of several modern Monte Carlo parton-shower generators and generator tunes, and a wide range of agreement with the data is observed, with $p$-values varying from $5 \times 10^{-4}$ to 0.98. These measurements complement similar measurements from $e^+e^-$ collider experiments in which the $b$-quarks originate from a color-singlet $Z/\gamma^*$.