Long-lived particles represent a well motivated approach for beyond-Standard Model (SM) physics searches. An interesting scenario at the LHC is the one in which light vector mediators (dark photons), weakly coupled to the SM photon, can be produced by an exotic decay of the SM Higgs boson and decay back to SM particles after travelling a macroscopic distance. This study presents a search for light, neutral long-lived particles decaying in collimated jet-like structures containing pairs of leptons or quarks. The search is performed on 139 fb$^{-1}$ of proton-proton collision data at $\sqrt{s}$ = 13 TeV collected by ATLAS during the Run-2. Both the gluon-gluon fusion and associated production with a W boson are considered for the Higgs production. Dark photon decays are identified among the overwhelming QCD and non-collision background using a selection involving dedicated triggers and deep-learning based classifiers. The results obtained are interpreted in the context of simplified long-lived particle models such as the Hidden Abelian Higgs Model and the Falkowski-Ruderman-Volansky-Zupan model.