Two-particle angular correlations explore particle production mechanisms and underlying event-wide phenomena present in the systems created in hadronic collisions. These correlations are examined as a function of rapidity and azimuthal-angle differences ($Δy, Δ\varphi$) for pairs of like- and unlike-sign pions, kaons, and (anti-)protons produced in pp collisions at $\sqrt{s}$ = 13 TeV, measured by the ALICE experiment. Two-particle correlation functions are provided, along with $Δy$ and $Δ\varphi$ projections, and are compared to Monte Carlo (MC) model predictions. For the first time, the measurement is performed as a function of the event's charged-particle density. The shapes of the correlation functions are studied in detail for each particle pair. Previous studies conducted for pp collisions at $\sqrt{s}$ = 7 TeV at ALICE have revealed an anticorrelation at small relative angles for baryon-baryon and antibaryon-antibaryon pairs, whose origin remains an open question. In this work, an additional approach is introduced to study the multiplicity dependence of the correlation functions in more detail and reveal qualitative differences in the underlying sources of correlations, such as quantum statistics, final-state interactions, and resonance decays. The puzzling near-side anticorrelation in baryon-baryon measurements is observed across all multiplicity classes and remains a challenge for models of particle production in pp collisions. Furthermore, the multiplicity dependence of the correlations between mesons provides an independent means to explore the sensitivity of current MC models to soft-QCD effects and hadronization dynamics. The presented measurements, together with the baryon results, enrich the experimental picture of two-particle correlations in pp collisions and serve as valuable input for ongoing theoretical developments.