I suppose because one can plug a mathematical formula that exactly yields pi into an equation and do mathematically meaningful things with it there, whereby you can't do that with the analog pi because it can only be as good as the actual probability experiments you carry out. Dropping your needle over and over is analogous to cranking out digits of pi.
No, the convergence properties are different. Analogously, consider tossing a coin and see if it cranks out the digits of 1/2.
Note that the coin or needle doesn't crank out anything; they're inanimate generally. Averaging over outcomes yields the aforesaid results.
Just for fun: pi/4 = 1 - 1/3 + 1/5 - 1/9 ....