The equation for capacitance is Q=CV or V=1CQ. I don't understand what is the physical meaning of this "C":
Does the charge in a system changes linearly with voltage under all circumstances?
This first part is the statement of the behavior of an "ideal" capacitor. Because it is an idealization, it is easy to characterize as a linear component whose behavior can be easily calculated. As a concept this gives it general applicability in many real-world situations, where important and valid results can be obtained by approximating the messy, imprecise, non-linear behaviors of things with idealized components given some constraints (such as limiting the system characterization to a small range of voltages and/or currents).
Non-ideal capacitors will not behave in this way, but this linear model of capacitance is so useful, and so generally applicable in modern, everyday electronic circuit theory, that the messy, general, nonlinear forms are not even considered.
As for the area of the plates... it seems intuitively obvious that the storage of charge in the "capacitor" must be directly proportional to some measure relating to "capacity". In a freely conducting medium, the electric field is uniform, so the only other parameter which can provide more capacity is the area of the plate: more charges require more area to spread out over, if the electric field is to remain uniformly at one value.
For the nature of the question, this seems like the best answer that can be afforded "intuitively" without going into great detail about Gauss's law and other aspects of electrostatics.