The Romanian physicist Proca formulated his famous Lagrangian to describe a hypothetical massive photon. From it we derive, as equations of motion, the relations that the electric and magnetic fields must obey (the analogous of Maxwell's equations for massive photons). We know that Einstein deduced the Lorentz transformations by raising the constancy of the speed of light to a principle. In this sense, giving a mass to the photon would mean going against this principle. However, we also know that transformations between inertial systems can be obtained based on the principles of relativity, isotropy and causality. So there are two types of groups (excluding space-time inversions) which simultaneously satisfy this principles: Galileo's and Lorentz. In the last $c$ appears as limit speed that cannot be exceeded and which, a priori, is not related to the speed of light. Despite this, however, we choose to consider the Lorentz transformations and to discard those of Galileo due to the fact that, since Maxwell's equations must be invariant, the speed of light must remain the same in all inertial reference systems. So, even in this way, Proca's hypothesis seems to make no sense. Can you explain to me why the Proca hypothesis makes sense? Furthermore, in this perspective, since the speed of light can no longer be the limiting speed in the Lorentz transformations, what meaning should we give to $c$?
Calling $c$, the parameter in Lorentz transformations, the speed of light is something of a misnomer. It should more properly be called speed of massless particles -- or as robphy suggested in the comments, '“maximum signal speed” (putting more emphasis on the causal structure).' Whether or not the photon is actually a massless particle is not relevant for the logic of special relativity.
In this scenario, $c$ would still be the maximum possible velocity at which information can propagate locally, i.e. in a piece of spacetime small enough to be considered Minkowskian, even if no information would actually travel with that speed.
The parameter $c$'s existence does not require light in Einstein's massless sense: that was just the heuristic he used to derive it. The relevant physical postulate that can be used instead of anything related to light directly is that there is an upper limit to the speed or, if you like, a lower bound on the per-distance latency, of transmission of information from one spatial point to another.