It's been a long time since I looked at this stuff, but I do recall there was at least a bit of research looking at e.g. cross-cultural differences in the factor structure and correlates of Five-Factor Model measurements. From what I remember, and a quick scan of some of the leading abstracts on a quick Google scholar search, the results are as you might expect: there are some similarities and parts that replicate, and other parts that don't [1].

I think when it comes to things like psychopathology though, there is not much research and/or similarity, especially relative to East Asian cultures (where the Western academic perspective is that there is/was generally a taboo on discussing feelings and things in the way we do in the West). The classic (maybe slightly offensive) example I remember here was "Western psychologization vs. Eastern somatization" [2].

The research in these areas is generally pretty poor. Meehl and Smedslund were actually intelligent and philosophically competent, deep thinkers, and so recognized the importance of conceptual analysis and semantics in psychology. Most contemporary social and personality psychology is quite shallow and incompetent by comparison.

Psychopathology research too has these days generally moved away from Meehl's careful taxometric approaches, with the bad consequences that complete mush concepts like "depression" are just accepted as good scientific concepts, despite pretty monstrous issues with their semantics and structure [3].

[1] https://scholar.google.ca/scholar?hl=en&as_sdt=0%2C5&q=five-...

[2] https://scholar.google.ca/scholar?hl=en&as_sdt=0%2C5&q=weste...

[3] https://www.sciencedirect.com/science/article/abs/pii/S01650...