I used to work at a company in which psychometrics were a major component. If you've ever taken a computer adaptive test, that test was based on a psychometric model.
For example in an adaptive test, question are trialed in order to determine how discriminating they are (i.e. which side of an ability estimate correctly answering places you), the likelihood of error in responding (maybe how confusing the item is or some other interference), likelihood of guessing, etc. It is essentially a statistical discipline with some grounding in psychology, but all of the pyschometricians I worked with considered themselves primarily statisticians.
I've never understood psychometry to be a theory about personality, though I suppose you could apply it to such a theory. Maybe its a case of the same term applied to two things?