I was just thinking the other day a birthday cake icon on ts for members birthdays would be fun, not necessary but fun. 
thats all super simple lol
I do not understand how you would use the TS birthday tag? Does someone have to manually add and remove the tag based on a database of birthdays?
I knew someone who had it linked to their website and would display the icon on TS for the day of the users birthday (I assume via a bot account or API) I have not personally set something like this up. I would not suggest that it be done manually by the admins, but have it based on the website accounts and the user is able to select if they want their birthday displayed or not in the user control panel
This would need to be developed from scratch. We have a very limited number of developers, and this has very little value for the time it would take to develop. I do not see it being done any time soon.