I agree that there is a lot of overlap between the kind of personalities who like FM and the kind who like Psych...as it should be, because the two fields have a lot of overlap. Many people with depression initially present to their family doctor (I also think that a high number of patients who present to the FM with vague somatic pains and that sort of thing often have a depression or anxiety component to their trouble). Meanwhile, on the psych units, it is very helpful for a psychiatrist to have some general medical knowledge since most of the severely mentally ill don't get consistent medical care and often will come to their psychiatrist with a myriad of complaints that aren't directly related to their mental illness.
Other reasons that I think FM people switch to Psych are because the lifestyle in residency tends to be better and it is the least physically demanding specialty (which I think is a big reason for why psych is attractive to career changers in general - if you are a surgeon who develops a health problem that makes operating difficult, switching to something like psych makes a lot of sense).
