The vector-boson production cross section for the Higgs boson decay in the $H\ensuremath{\rightarrow}W{W}^{*}\ensuremath{\rightarrow}e\ensuremath{\nu}\ensuremath{\mu}\ensuremath{\nu}$ channel is measured as a function of kinematic observables sensitive to the Higgs boson production and decay properties as well as integrated in a fiducial phase space. The analysis is performed using the proton-proton collision data collected by the ATLAS detector in Run 2 of the LHC at $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$ center-of-mass energy, corresponding to an integrated luminosity of $139\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. The different flavor final state is studied by selecting an electron and a muon originating from a pair of $W$ bosons and compatible with the Higgs boson decay. The data are corrected for the effects of detector inefficiency and resolution, and the measurements are compared with different state-of-the-art theoretical predictions. The differential cross sections are used to constrain anomalous interactions described by dimension-six operators in an effective field theory.