Why do we say "approximation" in the dipole approximation in spectroscopy? In the dipole approximation, the following relation holds:

$$
\hat{V} = -\hat{\mu} \cdot \vec{E}.
$$

When we say "approximation", I guess we want to point out that we are dealing with linear spectroscopy, where we have the polarizability of the molecule proportional to the electric susceptibility of a molecule:

$$
\vec{p}(\omega) = k\chi \vec{E}.
$$

If the wavelength of the light is not larger than the atom size, the dipole approximation is no longer a good approximation. This is probably also true for large molecules. I'm not sure if this has to do with large molecules no longer having symmetry, so transitions might be allowed which otherwise should be forbidden.

Is this correct?
As per your linked reference, it's an approximation because only the first term in the expansion (here, unity) is considered.

The issue with transitions that are allowed or forbidden arises as a result of using such an approximation: "Forbidden" transitions are not strictly forbidden, it's just that they are weak and, in the linear approximation of the transition dipole operator, are usually not observed.