Excellent question! There's two things going on here - properties ascribed to $\pi$ itself, and properties of the digits of $\pi$. In particular, properties of the number itself can't change when the base changes, (the ratio of circumference to diameter of a circle doesn't change if you lose a finger), so $\pi$ itself stays as it is. Properties like irrationality remain, since we know $\pi$ can't be written as a fraction of whole numbers, regardless of the base we're in. On the other hand, properties of the digits of $\pi$ can change! In other bases, $\pi$ wouldn't start off like the familiar $3.14\dots$, and interesting coincidences like the Feynman point won't exist any more.
On a deeper level, we don't know if the digits of $\pi$ appear "uniformly" in base 10, nor in any other base, related to the idea of a normal number. Talking about other bases specifically, the BBP algorithm gives a convenient way of computing the digits of $\pi$ in base 16, and as a spigot algorithm it doesn't rely on previous digits to find the next one, unlike most familiar algorithms for calculating $\pi$.