Currently there are approximately 13 mentioning birthday paradox and another 22 mentioning birthday problem (I guess there could be more). In most cases it is the main topic of the questions. Should we have separate tag for it? It is pretty important topic in probability so it may be worth having a tag.
Since there is no answers and not much comments, while the question and pro-tag comments received several up-votes I would consider creating the tag. Let's wait for few more days if anyone disagrees (please comment and/or down-vote this answer if you do). If there would be no disagreement about it I'll create the tag and add it to the already existing questions.
EDIT I started re-tagging from the highest up-votes questions. Check birthday-paradox tag.