I do not think that there is an official prize from Stake for birthday as I think that I will not get the tip if I did not show up in chat.
If it is an official promotion for birthday of the account then I do think there should be a direct gift without showing up in chat.
I have one absolutely since few years ago when I ranked up to Platinum 1.
I just remembered about my old VIP host (my first host who is resigned already IIRC), she asked me about my birth date for this thing.
I cant remember if I had received a birthday gift from her, but as I remember I got new host before my birthday.
I'm not complaining obviously as I think that Stake gave me many bonuses already so far.