  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 ($W^\pm W^\pm jj$) 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
$W^\pm W^\pm jj$ production are $2.92 \pm 0.22\, \text{(stat.)} \pm 0.19\,
\text{(syst.)}$ fb and $3.38 \pm 0.22\, \text{(stat.)} \pm 0.19\,
\text{(syst.)}$ 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 $H^{\pm\pm}$ 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 $H^{\pm\pm}$ mass near
450 GeV, with a global significance of 2.5 standard deviations.
