How are polarizabilities measured experimentally?
The polarizability of molecules determines how a molecule interacts with other molecules (intermolecular interactions and chemical reactions), how it interacts with electromagnetic radiation (optical properties), and how it interacts with electrical fields (electrical properties). Thus, there are multiple ways to compare the polarizability of two molecules or samples, and to quantify it.

[OP] Is the refractive index the most common method of experimentally measuring the polarizability?

That is hard to say without a systematic search of all the polarizability data. A starting point for such a search might be the polarizability data provided by NIST, which includes references that describe methods used. In general, there are electrical techniques (static electrical field or slowly oscillating) and optical techniques (electromagnetic field). Light scattering depends on the electric polarizability of the medium, so any technique that is based on light scattering (Raman spectroscopy, X-ray diffraction, refraction) in principle could be used to determine polarizability.

[OP] Can it be measured directly, or must be through a relationship such as those above?

The refractive index can be measured directly. The polarizability, while it has direct impact on molecular properties like the strength of dispersion forces, is determined from other measurements.

[OP] Does this relationship hold for oriented (anisotropic) polarizabilities, or only averaged (isotropic) ones?

As written in the question, the Lorentz-Lorenz equation is for the isotropic case. The literature linked to below also shows the more general equation.

[OP] The refractive index and the polarizability are frequency-dependent properties. How is the static (frequency-independent) polarizability measured? Or, is it even possible without extrapolation from multiple frequency-dependent data points?

The refractive index is frequency-dependent while the polarizability is not. You measure the refractive index at multiple wavelengths and extrapolate to infinitive wavelength. Alternatively, you can do a non-linear fit to the respective theoretical dispersion function.
Measurements of molecular polarizability are done at low concentrations in the gas phase, and yield the isotropic polarizability. Here is one paper that compares the experimental data to computational results for the iodine molecule. More complicated experiments involving polarized light are required to figure out the anisotropy, if any. For example, this paper from the same two labs compares the anisotropy of the polarizability of butadiene to computational results. An example of a quantumchemical computation of polarizability is here, for the hydrogen atom. In the solid phase with oriented molecules, the anisotropy of polarization is easier to measure (for example with cross-polarized filters as in the example below showing a plastic lens under mechanical stress):

Source: https://commons.wikimedia.org/wiki/File:Stress_analysis_of_plastic_lens.png