Please see question in title.
Why isn't philosophy considered to be a branch of mathematics?
Is study of anything not a branch of mathematics, vague and imprecise?
Philosophical arguments are made mathematical all the time. Its why you will see First Order Logic symbols thrown around on this Stack Exchange.
I think the big difference between mathematics and philosophy is that mathematics tends to start from something like a formal system, and see how much can be proven within it. Philosophy approaches the question of "what formal systems are right?" If a formal system proves something non-intuitive, Philosophers will immediately start studying the axioms of the formal system to see if they may be missing something. Philosophers admit more shades of "color" into their arguments than mathematicians can.
There are absolutely places where they blur. Consider the huge debate between ZF and ZFC in set theory. If you look at the wording choices, they are hard to distinguish from a philosophical debate. At the extreme, Proof theory and Model theory are almost more philosophical than mathematical. A read of the consequences of Tarski's definition of Truth is really almost entirely philosophy with only a pinch of Math for seasoning. Russel and Whitehead's "Principia Mathematica" could qualify as an attempt to make a religion out of pure math.
On the other side, philosophical issues like Xeno's paradox were deemed "insufficiently explained" until Calculus turned his philosophical argument into a mathematical argument. Mathematical inconsistency is often used to refute a position by using the strict meanings of FOL symbols like "AND" and "OR." These are used to force philosophers to identify where their arguments diverge from the world of mathematics. Turing's mathematical work strongly influences modern philosophy, especially with the development of AIs continuing as it is.
The real true difference is probably more linguistic than anything else. There is clearly a continuous gamut from pure philosophy to pure mathematics. However, we as humans tend to draw lines between terms because that makes them more meaningful. Mathematics tends to be colder than philosophy. Something is either formally proven, formally disproven, or an interesting outstanding problem. Philosophy is more open. Two philosophies may persist for centuries, not because nobody disproved one of the, but because people like both of them.
I think Philosophy is more willing to accept mathematics. However, mathematics avoids associating with philosophy except under well controlled conditions (such as those needed to support the richness that explodes from Formal Systems). This could account for the asymmetry in terminology you see.
Note: I say nothing about philosophers and mathematicians. The people in each group are as complicated as any person. I'm just talking about the topic described by the words.
I suppose it is true that there could be more activity towards taking philosophical arguments and trying to turn them into mathematics. The use of first order and modal logic in analytic philosophy is the most well-known attempt to make such a connection, and the rejection of much "continental philosophy" derives, to some extent, from the perceived impossibility of ever making any precise sense of it.
On the other hand, the mathematics traditionally considered in the spirit of analytic philosophy is typically rather simplistic compared to what practicing mathemacians care about. Maybe one might argue that making philosophical statements precise by means of first-order logic comes at the price of restricting oneself to a shallow fragment of philosophy.
However, there is a alternative proposal, which is little known, but deserves much more attention. Over the decades the mathematician William Lawvere had been proposing, more or less implicitly, that some key structures of just that Hegelian philosophy whose rejection is traditionally regarded as a prerequisite for having any precise argument at all, does have a very useful formalization -- not in first-order logic but in a richer categorical logic, in something one might call modal type theory. This subsumes modal first order logic, but is richer, in particular because there are adjunctions in this context which are not just Galois connections. Lawvere and others observed long ago that the concept of adjunctions formalizes aspects of duality that prevail through much of philosophy, see Lambek 82 "The Influence of Heraclitus on Modern Mathematics".
Lawvere has taken this to the point of concretely proposing formalization in categorical logic of the Hegelian concepts of unity of opposites and Aufhebung ("sublation"), concepts that are typically believed to defy all common sense. Turns out they do make precise sense, when regarded from a suitable perspective of mathematical foundations.
And this works rather well on both fronts: on the one hand, it does arguably capture some of the more mysterious sounding philosophical statements and illuminate them. Lawvere indicated some bits of Hegel's "Science of Logic" that are amenable to his formalization. I took the liberty of trying to elaborate on this a bit more, see on the nLab the entry Science of Logic. On the other hand, this does lead to mathematic that is of genuine interest as mathematics. In fact this has influenced some recent research that solved solved some actual open mathematical questions. I had commented on that in another thread at Have professional philosophers contributed to other fields in the last 20 years?
In conclusion, I am thinking that there is much potential to have renewed interaction between philosophers and mathematicians. It requires passing from first-order logic to type theory (and better yet to homotopy type theory) and it turns out to have something to say about just that kind of philosophy which is traditionally being rejected as unfit for precise argument.
why isn't philosophy just considered to be a branch of mathematics
OK I'm going to add another answer, on a different bent.
This is at bets like asking: why isn't Hindu philosophy just a linguistics that only asks about Sanskrit?
While historical and contemporary study of Hinduism is in Sanskrit, and there is something to be said for the two being inseparable, Hinduism is a body of thought which finds expression in Sanskrit but is irreducible to a scientific study of the language.
This may leave us asking well what is philosophy then, but that's nothing to be scared of.
Indeed, I personally believe that a reduction of philosophy to formal logic would be puzzling because philosophy would cease to exist. Pure logic can only answer questions about logic, not philosophical ones.
The overt intent of Logicism is to reduce all of philosophy to symbolic logic
Like Russell. These founding fathers of analytic philosophy
Dewey had described the writings... which sought to reduce philosophy to propositional logic, as "an affront to the common-sense world of action, appreciation, and affection".
In effect, that reduction is nowadays a straw man, I believe. I will try and find a quote!
why isn't philosophy just considered to be a branch of mathematics?
As Cort Ammon said in his excellent answer in the areas of logic and set theory there is considerable overlap to the extent that differences are mainly ones of emphasis.
However there are large areas of philosophy which are completely non-mathematical.
Examples would be:
1) The mind - body problem
2) Free will v determinism
3) Ethics (How should we behave? Is it possible to derive "ought" from "is"?)
4) The nature of being (What is the nature of the universe? Is there a god? What is man?)
5) Language games (Locke, Russell, Wittgenstein, Kripke, et al)
Is any study not a branch of mathematics vague and imprecise?
There have been many precise answers (not necessarily correct ;-) given to some of the questions above. For example (and paraphrasing for brevity):
"Knowledge is justified, true belief" (Plato)
"Ethical behavior is that which allows us to flourish" (Aristotle)
"We say A causes B if A always precedes B, A and B are proximate and there exists a convincing explanation for the connection" (Hume)
Actually, Even though they come from two areas, I can't see any difference in these two if they are reasoning. Mathematics may agree with this idea; but mathematicians won't. Notations and some other things curb them from transcending their subject.
Philosophical reasoning considers all the possible factors.
If a difference is essential, I shall give you one only for the essence: "One is for 'Yours' and the other is for 'You'--the eternal 'You' also."
Now you can find out the essence from my last statement itself. But don't forget the first statement.