When you create a custom date field, there is a set of check boxes for "Included date parts". You can turn off the year, month or day parts of the field. When I unchecked the year part, the contact data entry field correctly showed only the month and the day. However it won't let me save any value in the field because it always returns an invalid date error. Entering a default value for the field does not help.
With some experimenting, I found that any combination of month and day checked and unchecked choices are ok, but the year must be checked or it does not work.