This paper presents the measurement of fiducial and differential cross sections for both the inclusive and electroweak production of a same-sign W-boson pair in association with two jets (W$^{±}$W$^{±}$jj) using 139 fb$^{−1}$ of proton-proton collision data recorded at a centre-of-mass energy of $ \sqrt{s} $ = 13 TeV by the ATLAS detector at the Large Hadron Collider. The analysis is performed by selecting two same-charge leptons, electron or muon, and at least two jets with large invariant mass and a large rapidity difference. The measured fiducial cross sections for electroweak and inclusive W$^{±}$W$^{±}$jj production are 2.92 ± 0.22 (stat.) ± 0.19 (syst.) fb and 3.38 ± 0.22 (stat.) ± 0.19 (syst.) fb, respectively, in agreement with Standard Model predictions. The measurements are used to constrain anomalous quartic gauge couplings by extracting 95% confidence level intervals on dimension-8 operators. A search for doubly charged Higgs bosons H$^{±±}$ that are produced in vector-boson fusion processes and decay into a same-sign W boson pair is performed. The largest deviation from the Standard Model occurs for an H$^{±±}$ mass near 450 GeV, with a global significance of 2.5 standard deviations.[graphic not available: see fulltext]