We know that, under the simplifying assumptions that a year has 365 days (i.e., ignoring leap years) and that each day is equally likely to be a person's birthday, the probability that at least two people in a group of $n$ share the same birthday is

$$ 1-\frac{365}{365}\cdot\frac{364}{365}\cdot\frac{363}{365}\cdots\frac{365-(n-1)}{365}. $$

This probability increases nonlinearly with $n$. For example, it exceeds $50$% when $n=23$, is about $81$% when $n=35$, and about $99.4$% when $n=60$.

I have conducted this experiment on at least four different occasions, each involving around 30 participants. Every time, there was at least one pair of people sharing the same birthday.

This makes me wonder whether the assumption that birthdays are uniformly distributed over the 365 days is realistic. More generally, how can one formally test/assess the validity of the equal-likelihood assumption based on repeated observations of birthday matches?

PS: I do not have a formal background in statistics. (I have a math background) So, any rigorous explanation concerning solution of this question is greatly appreciated.