Question

How can I prove that the unit volt per meter and newton per coulomb refers to the same physical quantity? What's this quantity?

Answer 1

Both are equivalent as they express intensity of the electric field. Its SI units are newtons per coulomb (N⋅C−1) or, equivalently, volts per metre(V⋅m−1), which in terms of SI base units are kg⋅m⋅s−3⋅A−1.

There is analogy to the steepness of the slope (in gravitational field)  on each you place a mass. You can express it as a change of potential energy per unit of length or as a force acting on a unit mass.

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