The fragmentation of high-energy gluons at small opening angles is largely unconstrained by present measurements. Gluon splitting to $b$-quark pairs is a unique probe into the properties of gluon fragmentation because identified $b$-tagged jets provide a proxy for the quark daughters of the initial gluon. In this study, key differential distributions related to the $g\to bb¯$ process are measured using 33 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ of $s=13$ TeV $pp$ collision data recorded by the ATLAS experiment at the LHC in 2016. Jets constructed from charged-particle tracks, clustered with the anti-$kt$ jet algorithm with radius parameter $R=0.2$, are used to probe angular scales below the $R=0.4$ jet radius. The observables are unfolded to particle level in order to facilitate direct comparisons with predictions from present and future simulations. Multiple significant differences are observed between the data and parton shower Monte Carlo predictions, providing input to improve these predictions of the main source of background events in analyses involving boosted Higgs bosons decaying into $b$-quarks.