This academic degree still existed by the 29th century. Some of the bearers of this degree worked at the Temporal Integrity Commission. On the dedication plaque of the USS Relativity, R. Berman, P. Lauritson, R. Beltran, T. Russ, G. Wang, and J. Ryan were listed with this degree. (VOY: "Relativity", dedication plaque)
As a psychiatrist – a medical specialist, unlike a psychologist – Dr. Dehner most probably had an MD as well. (This would be consistent with the many people today who hold doctorates in both medicine and a related field.)
It seems likely that several of the non-medical doctors referenced on Star Trek had PhDs. For example, Dr. Kila Marr or Dr. Telek R'Mor were in professions (biology and astrophysics) that, at least on "our" 21st century Earth, award the PhD for their doctoral/terminal degrees. The same is likely true for other doctors who were clearly not physicians, such as Dr. Richard Daystrom.
Dr. Dehner's file actually misspells the term, using P.H.D. rather than the correct PH.D. (Philosophy being one word)