The increase of strange-particle yields relative to pions versus charged-particle multiplicity in proton-proton (pp) collisions at the LHC is usually described by microscopic or hydrodynamical models as a result of the increasing density of produced partons or strings and their interactions. Instead, we consider the multiple partonic interaction (MPI) picture originally developed in the context of the PYTHIA event generator. We find that strangeness enhancement in PYTHIA is hidden by a large excess of low-$p_{rm T}$ multi-strange baryons, which mainly results from the hadronization of $u$-quark, $d$-quark and gluon ($udg$) strings. Strange baryons produced in strings formed from parton showers initiated by strange quarks ($s$-fragmentation), however, describe well the spectral shapes of $Xi$ and $Omega$ baryons and their multiplicity dependence. Since the total particle yield contains contributions from soft and hard particle production, which cannot be experimentally separated, we argue that the correct description of the $p_{rm T}$-spectra is a minimum requirement for meaningful comparisons of multiplicity dependent yield measurements to MPI based calculations. We demonstrate that the $s$-fragmentation component describes the increase of average $p_{rm T}$ and yields with multiplicity seen in the data, including the approximate multiplicity scaling for different collision energies. When restricted to processes that reproduce the measured $p_{rm T}$-spectra, the MPI framework exhibits a smooth evolution from strictly proportional multiplicity scaling ($K_{rm S}^0$, $Lambda$, where the $udg$-hadronization component dominates) to linearity ($s$-fragmentation) and on to increasingly non-linear behavior ($c$-, $b$-quark and high-$p_{rm T}$ jet fragmentation), hence providing a unified approach for particle production in pp collisions.
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