It is a very basic question, but could not find it elsewhere and I think is the source of many of the misunderstandings that I have. My understanding is that in principle you can define a hierarchy of formulas of the form $\Pi^m_n$ and $\Sigma^m_n$, for any math theory, provided you use the appropriate language. But I am not sure if when I see those symbols in the literature they always refer to formulas of PA.
No, the symbols are also used elsewhere. For example indescribable cardinals do not refer to cardinals which cannot be described by formulas in the language of arithmetics.
Taken from Kanamori, The Higher Infinite:
In this context $Q$ is $\Sigma^m_n$ or $\Pi^m_n$. But the language itself is an arbitrary language containing two symbols, and certainly not over $\Bbb N$.
You can read more in the first section, (p. 5) where Kanamori discusses hierarchies of formulas.