What makes "Modal Logic" Logic? Why are symbols that stand for "necessary", for example, taken as symbols of Logic (of the same level of symbols that stand for "exists")? What are the limits that define Logic among other linguistic practices?
Why shouldn't it be a form of logic? I think that people are somewhat used to the idea of logic only being able to reason about the truth value of statements, but there are a lot of other things you might want to reason about. For example (and I'm oversimplifying a lot here):
- System state (dynamic logic)
- Time (temporal logic)
- Necessity/possibility (modal logic)
- Justification (one interpretation of constructive logic) - i.e. if the premises are justified, must the conclusion also be justified?
These are all forms of logic because you can make inferences from a premise to a conclusion while "preserving" some property of the premises. For example, classical logic is "truth preserving," meaning that if the premises are true and the proof is valid the conclusion will also be true. Constructive logic (in one interpretation at least) is justification preserving - meaning that if the premises are justified and the proof is valid, the conclusion will also be justified.
Granted, the distinction between logic and other fields becomes a little blurry at points. For example, there's a correspondence between constructive logic and computer programs.
Here's an argument from Aristotle which I'm paraphrasing:
If we ought to philosophise, we ought to philosophise
If we ought not to philosophise, then we ought to philosophise
in any case, we ought to philosophise...
This could be formalised by deontic modal logic.
Logic argues from valid premises, by a valid argument, to valid conclusions; this does not mean that modality need be ignored, and in fact historically it wasn't; peripatetic logic studied modality.
because the symbols do not in fact stand for "necessarily" (let's use  for this) and "possibly" (using <>). logic is purely formal; if these symbols had predefined substantial meanings they would not qualify as logical constants. their meanings are given by the formal rules of the language, not by the informal English meanings that inspired their invention.
so: P <-> ~<> ~P
informally, necessarily P iff not (possibly not P). But you can replace "necessarily" and "possibly" with anything, since this sentence just states a rule of the game. (try replacing them with e.g. "formally" and "informally", resp., or even "angrily" and "happily".)
Traditionally, logic is the part of the study of language that considers which statements conserve truth, and which do not. That can involve reasoning about obligations or possibilities and predictions of the future as well as present-time facts, as long as the question is formally about the correctness of deduction instead of the actual meaning of the terms involved. As pointed out, by others here, the Organon covers necessity as a category of logic.
From that point of view, modal logic and other related variants are clearly more a part of logic than set theory, which broadens the definition of the field to look within properties and definitions and model parts of their potential domains of references. This is at least semiotics, if not semantics, but we call it symbolic logic, and it has become central to the study of the discipline.
In reality, there is a continuum from syntax to logic to semiotics to semantics to ontology and epistemology that is very hard to divide up in a reasonable way, and our current view of each of these disciplines spreads across the boundaries somewhat into its neighboring disciplines, because it is hard to maintain a footing and be clear if you remove all of the context.
Quine took modal logic to invite translation issues, inference differences, and non-standard and seemingly unnecessary (outside of modal logic, for Quine) ontological commitments. For more on this see: http://johnmacfarlane.net/142/against-quantified-modal-logic.pdf