Is there a meaningful example of probability of $\frac1\pi$?

A large portion of combinatorics cases have probabilities of $\frac1e$.

Secretary problem is one of such examples. Excluding trivial cases (a variable has a uniform distribution over $(0,\pi)$ - what is the probability for the value to be below $1$?). I can't recall any example where $\frac1\pi$ would be a solution to a probability problem.

Are there "meaningful" probability theory case with probability approaching $\frac1\pi$?

Yes there is!

Here is an example, called the Buffon's needle.

Launch a match of length $1$ on a floor with parallel lines spaced to each other by $2$ units, then the probability that a match crosses a line is

$$\frac 1\pi.$$

You can have all the details of the proof here if you like.

$\qquad\qquad\qquad\qquad\quad $

More generally, if your match (or needle, it's all the same) have a length $a$, and the lines are spaced to each other by $\ell$ units, then the probability that a match crosses a line is

$$\frac {2a}{\pi \ell}.$$

This doesn't addresses the question as asked, but I think is an interesting example in the spirit of what you asked:

Pick $N$ to be an integer. Now, calculate $p_N$ to be the probability that two random numbers $1 \leq m,n \leq N$ are relatively prime.

Then, $$\lim_{N \to \infty} p_N =\frac{6}{\pi^2}$$

There are quite a few geometric probabilities related problems, involving $\pi$. The simplest I can think of, imagine this being a darts board.

The probability of hitting the square, assuming the shot didn't miss the board, is $\frac{2}{\pi}$.

If the radius of the circle is $r$, then one side of the square is $r\sqrt{2}$ and $p=\frac{S_{square}}{S_{circle}}=\frac{2 r^2}{\pi r^2}=\frac{2}{\pi}$