IQ tests aren't illegal by themselves, but it's up to the employer to show that IQ is an important part of the job. E.g, leetcode tests in FAANG interviews are defacto IQ tests since interviewers are told to evaluate the candidate for "general cognitive ability" when asking leetcode questions, but they are still used for FAANG interviews because the companies realized that they correlate to job performance. I don't know of any formal MBTI studies related to engineers, but my anecdotal data suggests that xNTx MBTIs are disproportionately present in SV. So there might be a useful reason to ask about MBTI in an interview, but it would be bad to decline a candidate based on their MBTI unless there was hard data suggesting it affected job performance.
MBTI’s Wikipedia page says it’s “criticized as pseudoscience and is not widely endorsed by academic researchers in the field. The indicator exhibits significant scientific (psychometric) deficiencies”, so I think hoping for formal studies by anyone other than the company that hypes it is unlikely
It is very well-studied in psychology. To say that most academic researchers don't "endorse" it is silly— it's not the job of a researcher to endorse a particular survey instrument, but rather to dispassionately study the mind using the best instruments available. That said, the Big-5 personality scale is certainly a more standard survey instrument in psychology, and it really doesn't matter too much which survey instrument you use, because they all correlate together. They just differ upon where you draw the coordinate axes in the space of personality; but they're all measuring essentially the same space of personality.
