I use Skype on my Windows phone as well as on my tablet and various other devices. Only the Windows Phone variant lights up with a notification when it is someone's birthday. This is just not information I need about most of the people on my Skype list. Either I know (my mother) or I don't care (a business contact I had one call with once and have left in my contacts list.) The handful of people I speak to regularly but don't already know their birthdays is not enough for me to want this feature.
But I can't turn it off! I suspect I turned it off on my desktop and Metro versions of Skype, since they don't tell me. I don't see it as a setting on the phone though. Can it be turned off?