= Birthday square-root scale
{title2=$n\asymp\sqrt m$}
The <union bound> gives $P_{n,m}\leq\min(1,n(n-1)/(2m))$. Using $1-x\leq e^{-x}$ on the distinct-label product gives
$$
1-e^{-n(n-1)/(2m)}\leq P_{n,m}\leq\min\left(1,\frac{n(n-1)}{2m}\right).
$$
The lower bound reaches at least one half when $n(n-1)/(2m)\geq\log2$. The simple approximation $n\approx\sqrt{2m\log2}$ predicts a group size near 23 for $m=365$. It is the quadratic number of pairs that makes the <square root> appear.
For a precise limiting statement, let $m\to\infty$ and $n/\sqrt m\to c\geq0$. Expanding the <logarithm> of the complement yields
$$
\log(1-P_{n,m})=-\frac{n(n-1)}{2m}+O\left(\frac{n^3}{m^2}\right),\qquad
P_{n,m}\longrightarrow1-e^{-c^2/2}.
$$
On this scale the remainder tends to zero. Thus a fixed substantial collision probability appears after order $\sqrt m$ draws. For $m=2^b$ uniform output labels, the corresponding scale is order $2^{b/2}$; <matching one specified birthday> instead involves a number of draws of order $m$.
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