A search for strongly produced supersymmetric particles using signatures involving multiple energetic jets and either two isolated same-sign leptons ($e$ or $\mu$), or at least three isolated leptons, is presented. The analysis relies on the identification of $b$-jets and high missing transverse momentum to achieve good sensitivity. A data sample of proton--proton collisions at $\sqrt{s}= 13$ TeV recorded with the ATLAS detector at the Large Hadron Collider in 2015 and 2016, corresponding to a total integrated luminosity of 36.1 fb$^{-1}$, is used for the search. No significant excess over the Standard Model prediction is observed. The results are interpreted in several simplified supersymmetric models featuring $R$-parity conservation or $R$-parity violation, extending the exclusion limits from previous searches. In models considering gluino pair production, gluino masses are excluded up to 1.87 TeV at 95% confidence level. When bottom squarks are pair-produced and decay to a chargino and a top quark, models with bottom squark masses below 700 GeV and light neutralinos are excluded at 95% confidence level. In addition, model-independent limits are set on a possible contribution of new phenomena to the signal region yields.