fft – Can we say that \$S_{ij}(f_k)\$ as measured by VNA at frequency \$f_k\$ is exactly the frequency response to that frequency, i.e. \$H(f_k)\$?

The answer is ‘yes, but’.

S parameters are defined to be measured in a specific system impedance. If your definition of H(f_k) includes those terminations, then yes, S₂₁(f_k) = H(f_k)

S₁₁ doesn’t tend to have a corresponding H

You appear to be making a distinction between H(f_k) and H(f), one being continuous, the other being discrete. This will make formal changes to the definitions.

If you want to use the Fourrier Transform, then frequency response should be defined from DC upwards. S parameters are often measured on bandlimited parts. There are approaches to relate the two, but they have to be used explicitly.