What are some possible reasons that projective personality tests are still used, even though their psychometric properties are fairly poor?
It is true that projective personality tests, as well as many other types of personality tests, are not particularly reliable or valid but are still being used. The most well known test of this type is the Rorschach Inkblot Test, but there are others, like the TAT (Thematic Apperception Test), which are commonly used to identify "underlying and unconscious motivations or attitudes." One of the most commonly used projective test is the sentence completion test. These continue in use because they are accepted as a means of delving into difficult to state "conscious and/or unconscious attitudes, personality characteristics, motivations, and beliefs."
First, they are often used as a kind of introductory exercise in order for therapists to get more familiar with their clients. In this case, the tests are a familiar (we have all seen them in movies and other places) and useful tool rather than an accurate diagnostic instrument.
Second, even if they are used as a diagnostic tool, in a purely therapeutic setting, the results probably do not deny the clients any rights or opportunities. For example, the results, in this specific setting, are not likely to impact employment or educational opportunities.
Third, if these tests are used in a therapeutic setting, the consequences of a misdiagnosis seem minimal, though perhaps an individual client who gets improper medication might disagree.
Because some of these projective tests rely so much on uncontrollable factors (such as the examiner's tone, the setting, the attitudes of both participants, the examinee's level of honesty, and examiner scoring), they are not generally recognized as effective diagnostic tools. Despite that, they do have some limited uses.
One common use is in employment screening in which the sentence completion test is often used to uncover conflicting beliefs that would exclude applicants who demonstrate decidedly antithetical values (motivation, belief etc) that would conflict with the employer's ethics or objectives.