A measurement of the inclusive jet production in proton-proton collisions at the LHC at $\sqrt{s}=13$~TeV is presented. The double-differential cross sections are measured as a function of the jet transverse momentum $p_\mathrm{T}$ and the absolute jet rapidity $|y|$. The anti-$k_\mathrm{T}$ clustering algorithm is used with distance parameter of 0.4 (0.7) in a phase space region with jet $p_\mathrm{T}$ from 97 GeV up to 3.1 TeV and $|y|<2.0$. Data collected with the CMS detector are used, corresponding to an integrated luminosity of $36.3~\mathrm{fb}^{-1}$ ($33.5 \mathrm{fb}^{-1}$). The measurement is used in a comprehensive QCD analysis at next-to-next-to-leading order, which results in significant improvement in the accuracy of the parton distributions in the proton. Simultaneously, the value of the strong coupling constant at the Z boson mass is extracted as $\alpha_S (m_\mathrm{Z})=0.1170 \pm 0.0019$. For the first time, these data are used in a standard model effective field theory analysis at next-to-leading order, where parton distributions and the QCD parameters are extracted simultaneously with imposed constraints on the Wilson coefficient $c_1$ of 4-quark contact interactions.