The striking similarities that have been observed between high-multiplicity proton-proton (pp) collisions and heavy-ion collisions can be explored through multiplicity-differential measurements of identified hadrons in pp collisions. With these measurements, it is possible to study mechanisms such as collective flow that determine the shapes of hadron transverse momentum (p_T) spectra, to search for possible modifications of the yields of short-lived hadronic resonances due to scattering effects in an extended hadron-gas phase, and to investigate different explanations provided by phenomenological models for enhancement of strangeness production with increasing multiplicity. In this paper, these topics are addressed through measurements of the K^⁎(892)⁰ and ϕ(1020) mesons at midrapidity in pp collisions at s= 13 TeV as a function of the charged-particle multiplicity. The results include the p_T spectra, p_T-integrated yields, mean transverse momenta, and the ratios of the yields of these resonances to those of longer-lived hadrons. Comparisons with results from other collision systems and energies, as well as predictions from phenomenological models, are also discussed.