  We study the scenario of dark photon Dark Matter where the mass is generated
through the Higgs mechanism rather than the constant Stueckelberg mass. In this
construction the dark sector contains necessarily extra degrees of freedom and
interactions that lead to non-trivial dynamics including thermalization, phase
transitions, cosmic string production. As a consequence the predictions of
Stueckelberg theories are vastly modified, strongly depending on the couplings
to curvature and on the scale of inflation $H_I$ compared to the scale $f$ of
spontaneous symmetry breaking. We find in particular that only in extreme
regions of parameter space the phenomenology of Stueckelberg dark photon is
reproduced. These scenarios are strongly constrained by isocurvature
perturbations unless the dark sector is approximately Weyl invariant.
