I understand spontaneous symmetry breaking and the Higgs mechanism pretty well in the context of high energy physics (Relativistic QFT). Although I do also understand these phenomena in condensed matter physics (Non-Relativistic QFT), I might ignore some of the details.
For the sake of simplicity, I will talk about an abelian U(1) (global or gauge) symmetry. My understanding is the following:
A complex Klein-Gordon field (charged or uncharged under a gauge interaction depending on whether we are talking about a SSB or a Higgs mechanism) takes a constant VEV, so the vacuum is full of this field. The fluctuations around the VEV are described by another complex field (which only differs from the first one by the the VEV). In the RQFT case, the latter complex field may be discomposed in a component in the direction of the U(1) symmetry ("the bottom of the Mexican hat") and another radial component, that is, a goldstone boson (a massless propagating particle in the case of a SSB and a longitudinal mode of the gauge boson in the Higgs mechanism) and a massive independent particle (a sigma meson in a linear sigma-model -- where there is SSB -- and the Higgs particle in the case of the Higgs mechanism). These two degrees of freedom -- the massless goldstone (or the longitudinal polarization of the gauge boson) and the radial massive particle -- are in correspondence with the two degrees of freedom associated to the original complex Klein-Gordon field (the would-be particle and antiparticle).
The situation is different in non-relativistic QFT (condensed matter). Here, a complex Schrödinger field represents a single degree of freedom, as the Lagrangian only contains a time derivative. This is the Schrödinger field connected with the atoms of a superfluid (SSB) or with the Cooper pairs of a superconductor (Higgs mechanism). In theses cases, as far as I understand, there is a massless excitation (Goldstone boson) so-called phonon in the former case and an additional longitudinal polarization of the photon in the superconductor case. But there are no radial massive excitations (what would-be a sigma meson or a Higgs boson, respectively) in any case. The reason is clear: the Schrödinger field leads to a single degree of freedom instead of the two degrees of freedom due to the a complex Klein-Gordon field.
My question is the following. If, as I stated, there is no possibility of a Higgs-like excitation in a superconductor just because of the conservation of the number of degrees of freedom, what is a "Higgs mode" in superconductors and why is it called this way?