Optional year on Date of Birth demographics field: is it possible?
This question led me to re-investigate a 'feature' that was added quite a few years back to allow sites to set different date formats for categories of date fields. In 4.6, you access this via Administer > Customize > Date Preferences. There is a 'category' for Birth Date which "should" allow you to select alternative formats including mm/dd OR yyyy.

Unfortunately, testing in 4.4, 4.5, and 4.6 shows that there's a long-standing regression for this functionality (which I guess was rarely used). The Birth Date input field 'appears' to accept dates in the alternate format, but they aren't saved properly in any of the current versions.

Pretty sure fixing this will involve mods to the schema, processing code and the date input widget - so non-trivial. If someone is interested in working on a fix for 4.7 - patches definitely welcome.