For example, $\epsilon$ and $\delta$ are used in Real Analysis for proving limits... $\phi$, $\nu$ and $\mu$ were introduced to me in the context of number theory... and then $\pi$ is used for purposes other than its canonical $3.14...$ value, such as to represent the equation of a plane, or as a function name.
If we all use the same letter for the same thing, then we can skip the explanation. Imagine if you had to preface every post with "Let $π$ be the ratio of a circle's circumference to its diameter." When you approach an $\epsilon$-$\delta$ proof, if the author uses the familiar letters in their familiar roles then the reader familiar with the proof technique is already halfway there understanding the proof. That explains the consistency.
As for why any particular letter is used, such as why do we use $\pi$ instead of $\rho$ or $\psi$, the short answer is "tradition". Folk etymology abounds, but it takes serious scholarship in the history of mathematics to get real answers. For new coinages, the limiting factor is which letters aren't already being used. That's why nowadays new symbols sometimes come from exotic alphabets.
It's worth noting that a letter will often suggest its own meaning. For example, in most languages in which mathematics is written, the word for plane has a prominent P sound, and the Greek letter for that is $\pi$. Maybe that's why so many authors will use the symbol $\pi$ to refer to a plane.
If your question is, why is the Greek alphabet used instead of any other, the answer is that until relatively recently in history, all educated people were expected to be able to read at least a little Greek. Therefore aside from the Latin alphabet, the Greek alphabet was the next-most easily recognized alphabet.