Sometimes people will give their birthday but not their year of birth. As far as I can tell, it's not possible to enter these (if it is, I haven't figured out how).
For example, the date of birth of https://www.wikidata.org/wiki/Q11194654 is the 27th of August according to http://cleanero.com/profile.html but no year is given. Most of the pages in https://ja.wikipedia.org/wiki/Category:%E7%94%9F%E5%B9%B4%E9%9D%9E%E5%85%AC%E8%A1%A8 ("birth year unknown") also only have the month and day.
It seems that ISO 8601 supports these by using a hyphen to represent the unknown part, e.g. --02-03 for the 3rd of February in an unknown year.
Note that this is similar to, but distinct from, recurring dates.