The production of $Z$ bosons in association with a high-energy photon ($Z\gamma$ production) is studied in the neutrino decay channel of the $Z$ boson using $pp$ collisions at $s$ = 13 TeV. The analysis uses a data sample with an integrated luminosity of 36.1 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ collected by the ATLAS detector at the LHC in 2015 and 2016. Candidate $Z\gamma$ events with invisible decays of the $Z$ boson are selected by requiring significant transverse momentum ($pT$) of the dineutrino system in conjunction with a single isolated photon with large transverse energy ($ET$). The rate of $Z\gamma$ production is measured as a function of photon $ET$, dineutrino system $pT$ and jet multiplicity. Evidence of anomalous triple gauge-boson couplings is sought in $Z\gamma$ production with photon $ET$ greater than 600 GeV. No excess is observed relative to the Standard Model expectation, and upper limits are set on the strength of $ZZ\gamma$ and $Z\gamma \gamma$ couplings.