I don't think I can answer this question precisely, but I would like to say some things.
Since this question was asked, much has changed in the field. We have the following known sources that may interest any reader passing by (since probably by now the OP is an expert in the subject, I hope this answer can be useful):
With all that in mind I just want to make some small comments:
There are experimental implementations but no arguably loophole-free
It is a matter of great debate what notion of contextuality is optimal, moreover, every notion has its peculiarities and the search is to a broad notion that is experimentally robust and without too many hidden assumptions. Many experiments have been performed and I don't know any groups that are currently performing experiments, but I know that every group searches for an experimental implementation of contextuality that is arguably loophole-free.
As far as I know, probably, groups are working with experimental data from experiments performed in Latin America and USA/Europe trying to grasp CbD from data tables. Groups are trying to perform experiments in quantum contextuality to apply in random number generation. Groups are proposing experimental implementations of violations of contextuality/nonlocality inequalities using quantum linear optics. There probably are groups trying to implement new loophole-free experimental tests of generalized contextuality (but I only know about the theoretical developments). Since this is an ongoing active research field, different groups do not easily share their experiments much before having good enough results.
As there is not a specific notion of contextuality that is considered the best (and likely there will never be, I think it more possible that all different notions will be better applied in different circumstances) different notions have different loopholes to worry about. In generalized contextuality, for example, an important loophole is the fact that ideal operational equivalences are usually not present in real-life experiments, and secondary procedures must be described (see the experimental references I have mentioned already).
See for example this presentation here discussing loophole-free tests in the framework of Abramsky-Brandenburger contextuality.
Is "quantum contextuality" generally held to be a uniquely quantum phenomenon, or is it meant as an analog to some sort of non-quantum contextuality?
Contextuality is not a quantum phenomenon. Contextuality is a phenomenon that is satisfied by quantum theory. This is something that is made clear by every description of contextuality, in particular, almost every presentation by Samson Abramsky has the maxima that contextuality refers to the fact that probabilistic data may present
"local consistency but global inconsistency"
and to me, this description makes clear that there is nothing quantum about contextuality, but there is something contextual about quantum! And we know this for more than 50 years now, that quantum theory cannot be explained by contextual empirical models (sticking to sheaf-approach terminology in this sentence at least).
When we describe the notion of Generalized Contextuality, we treat it in the framework of operational-probabilistic theories, or also in the GPT framework, which has quantum theory as a particular instance.
In the CbD approach, this is also noticed as the fact that contextuality is a property of random variables, and not of quantum theory itself. The most relevant aspect of this discussion corresponds to the fact that exist post-quantum correlations.
Adding to the discussion in the comment section: we know that every noncontextual empirical model has a quantum realization, and with that I want to say that given any scenario, the probabilities arising in this scenario (depends on the scenario, non-local scenarios for example have probabilities of the form $p(ab|xy)$ with $a,b$ possible results for $x,y$ measurements in each party side) have a description in terms of the Born rule (therefore, $p(ab|xy) = Tr(\rho E_a^x \otimes E_b^y)$).
With contextuality not being a quantum aspect, I mean that we don't need quantum theory to infer if a given data-table is contextual or not. Quantum theory is a specific type of theory that cannot be explained only by noncontextual data-tables. Less known is that this property is true both for KS contextuality and for Generalized contextuality as well.