This paper presents measurements of top-antitop quark pair (t (t) over bar) production in association with additional b-jets. The analysis utilises 140 fb(-1) of proton-proton collision data collected with the ATLAS detector at the Large Hadron Collider at a centre-of-mass energy of 13TeV. Fiducial cross-sections are extracted in a final state featuring one electron and one muon, with at least three or four b-jets. Results are presented at the particle level for both integrated cross-sections and normalised differential cross-sections, as functions of global event properties, jet kinematics, and b-jet pair properties. Observable quantities characterising b-jets originating from the top quark decay and additional b-jets are also measured at the particle level, after correcting for detector effects. The measured integrated fiducial cross-sections are consistent with t (t) over barb (b) over bar predictions from various next-to-leading-order matrix element calculations matched to a parton shower within the uncertainties of the predictions. State-of-the-art theoretical predictions are compared with the differential measurements; none of them simultaneously describes all observables. Differences between any two predictions are smaller than the measurement uncertainties for most observables.