
WJM's date of birth is real and here's my theory.
The only thing we know that isn't true is his name, which even his closest people call him by on a regular day. Also, his date of birth fits perfectly before Louis'. (These things shouldn't be taken at face value like how he like smile and dislikes tears that would mean that he likes to bring joy to others and make their suffering evaporate and not that he is allergic to teardrops like how Sherlock is to women.)
And thus I conclusion that all the information given in their character bio is true if one reads between the lines.