Measurements of the Standard Model Higgs boson decaying into a $b\bar{b}$ pair and produced in association with a $W$ or $Z$ boson decaying into leptons, using proton-proton collision data collected between 2015 and 2018 by the ATLAS detector, are presented. The measurements use data delivered by the Large Hadron Collider at a centre-of-mass energy of $\sqrt{s} = 13$ TeV, corresponding to an integrated luminosity of 139 $fb^{-1}$. Cross-sections of associated production of a Higgs boson decaying into bottom quark pairs with an electroweak gauge boson, $W$ or $Z$, decaying into leptons are measured as a function of the gauge boson transverse momentum in kinematic fiducial volumes corresponding to the stage-1.2 simplified template cross-sections (STXS) scheme. Only the central kinematic region with an absolute rapidity of the Higgs boson $|y_{H}|$ of less than 2.5 is considered. The cross-section measurements are all consistent with the Standard Model expectations, and the total uncertainties vary from 30% in the high gauge boson transverse momentum regions to 85% in the low regions. Limits are subsequently set on the Wilson coefficients of a number of dim-6 operators in the Standard Model Effective Field Theory (SMEFT) which modify the $WH$ and $ZH$ processes as well as the Higgs boson decay to $b\bar{b}$. These limits are also expressed in terms of constraints on an orthogonal set of linear combinations of operators to which this analysis is sensitive.