  Two Higgs doublet models are attractive scenarios for physics beyond the
Standard Model. In particular, lepton-specific manifestations remain contenders
to explain the observed discrepancy between the anomalous magnetic moment of
the muon $a_\mu$ predicted within the Standard Model and recent observations at
Fermilab and BNL. Dominant uncertainties that affect $a_\mu$ have motivated the
MUonE experiment to access the hadronic vacuum polarisation contribution that
impacts $a_\mu$ via elastic muon-electron scattering. In this work, we contrast
the high precision that is achievable within the MUonE context with constraints
from flavour physics, precision electroweak constraints and LHC searches as
well as their extrapolations for a range of two Higgs doublet models with a
softly broken $\mathbb{Z}_2$ symmetry. We find that the sensitivity of MUonE
does not extend beyond the parameter regions that are already excluded by other
constraints. MUonE will therefore provide a detailed measurement of the
hadronic vacuum polarisation contribution which then transparently informs
$a_\mu$ interpretations in 2HDMs without modifications of correlations from
beyond the Standard Model interactions. In passing we extend earlier results of
LHC and flavour projections to lepton-specific 2HDM (Types X and Y) scenarios,
and comment on the possibility of modifying the value of the W-boson mass; we
briefly discuss the implications for a strong first-order electroweak phase
transition for these models.
