That's one of the examples he goes into in his talk, at a high level anyway. I'd love to find a write-up of someone who has done the detailed calculations of how likely alignments and shape occurrences are.

Another thing that would be interesting is to look at the effect of non-uniformly distributed birthdays. For example, the day that's nine months after valentine's day or christmas might (?) have a slightly higher number of births than an average day. Then you could look at what kind of an effect this would have on the probability of a common birthday as a function of group size.