So lets do away with months/days and assume everyone has a birthday $X_k$ which corresponds to a number from $1$ to $365$, uniformly distributed
In a group of $n$ people, let $M=\max (X_k)$ be the last birthday of the year and let $N=|\{X_1,X_2,\dots, X_n\}|$ be the number of distinct birthdays.
Now it seems obvious that $E[M]\approx \frac{365n}{n+1}$ and maybe $E[M^2] \approx \frac{365^2n}{n+2}$ by approximating as a uniform r.v.
As for $N$, we could get $E[N]=n\left(1-\left(\frac{364}{365}\right)^n\right)$, and maybe $E[N^2]$ by a similar argument to the elevator question...
My real question is: can we find a way to calculate $E[MN]$? Since $M$ and $N$ are dependent, I cannot find a nice way to express their joint distribution and therefore this expectation escapes me