The most intuitive explanation I've seen, although a bit abstract, is that Fourier transforms, and the other Fourier-like transforms including the s and z transforms, are just about changing the system of coordinates of the data vector using the transform's generative vectors.
Indeed we can easily see that the definition of all those transforms take the form of the sum of a scalar product that computes the projection of the data vector unto the kth dimension in the transformation (e.g. frequency) space.
This is the interpretation that leads you to imagine what a fractional Fourier transform might be: a rotation other than 90 degrees in the vector space you described. Interesting, if esoteric.
I was wondering recently whether some kind of 45° rotation transform could be used to represent equally well both "sides" of Heisenberg's unsharpness principle ?
Indeed we can easily see that the definition of all those transforms take the form of the sum of a scalar product that computes the projection of the data vector unto the kth dimension in the transformation (e.g. frequency) space.