At the University of Georgia we have a program (which is not identical to a "department", but is close enough so that the distinction has always eluded me) in Mathematics Education and thus we have undergraduate mathematics education majors and also mathematics education graduate students (both master's and PhD).
There are certainly close ties with both the mathematics department and other education disciplines -- e.g. the mathematics courses I teach have a substantial population of mathematics education majors -- but such undergraduate majors also take plenty of courses with the name "Math Education XYZW". These courses are split into "content" and "pedagogy" courses. This distinction was very hard for me to wrap my mind around (it literally took me a few years to do so, although obviously I was not working very consistently on it!): see for instance
this page, and please read carefully: the courses that they list as Content are actually in the math department (one or two of them are taken mostly by math education majors, but many of the others are also required courses for undergraduate math majors). Rather the distinction between "content" and "pedagogy" -- which are the terms used by UGA students and faculty who talk to me about this -- is a distinction being made between two types of courses in the longish list of Professional Education courses. Thus e.g. compare
Course Title: Connections in Secondary Mathematics II
Course Description: Exploration of secondary mathematics topics related to number and measurement with an explicit focus on reasoning that connects critical topics of secondary mathematics to one another and to problem situations. Sample topics include proportional reasoning, number theory, and probability.
Course Title: Teaching Secondary School Mathematics I
Course Description: Introductory ideas about mathematics education, including current mathematics standards and policy documents, learning theories, and teaching strategies. Students will explore how secondary students think about and learn mathematics, examine how to select and modify tasks, use appropriate technology, and apply their learning in an accompanying field experience.
This seems to give rather strong evidence that the answer to the OP's first question is yes: Math Education is rapidly becoming distinct enough from Education in general to count as its own discipline. (Obviously there remain many connections and commonalities between Math Education and other kinds of Education, just as virtually any academic field overlaps significantly with others.) In particular, yes, math education students learn math-specific teaching methods. This is indirect, though strong, evidence that there are differences between the teaching methods of various subjects. But my other point is something that is not explicitly in the OP's question: more than just teaching methods, techniques or ability, there is actually additional content that math teachers learn and that other teachers (and students of mathematics who are not intending to teach!) do not.
Let me also introduce you to my colleague Sybilla Beckmann. Beckmann (who was trained as an arithmetic geometer and holds a faculty position in the UGA mathematics department; her office is next door to mine) is truly* one of the very top American experts in the field of mathematical education of teachers. (Beckmann is also largely responsible for my awareness and understanding of the issues presented above. In fact I will contact her and ask her to look over this response to make sure I have gotten things right.) One of her initiatives over the last few years has been to promote the idea of an explicitly identified mathematics teaching community. In this regard, please see this article and this website. Also, Beckmann writes on her webpage "Longer term, we plan for this project to include an electronic self-organizing journal."
tl;dr: Yes, this is definitely a thing. It is a thing which has grown in recent years and is liable to continue to grow in the near future...and everyone seems to agree that we want/need it to grow.
*: So much so that I need not justify it here: just search the web for her and you'll see it right away.
Added: My colleague Sybilla Beckmann took time out of her busy workday [on Saturday!] to quickly look over what I wrote above. She pronounced it "basically accurate" and went on to add the following:
Teaching methods in math are definitely different from other disciplines. Work of Lee Shulman and Deborah Ball on mathematical knowledge for teaching has been transformative for the field in that regard. Some sources to refer people to: the CBMS Mathematical Education of Teachers II on the CBMS website. It refers to various other sources. Math education is a separate discipline with a large body of research amassed over the last 30 - 40 years. It connects to other education research but is its own separate field. At UGA, the math ed program is within the department of mathematics and science education (it used to be a separate math ed department but was joined for administrative reasons some years ago).
I added links to the wikipedia articles on Shulman and Ball, and I recommend at least skimming these. Shulman is responsible for the idea of pedagogical content knowledge, a concept which is rather slippery at first [or at least, it was to me] but really seems to lie at the heart of an answer to the OP's question: it is precisely the material that you need to know as a mathematics teacher that you do not learn in your mathematics courses and cannot learn in non-math specific education courses. To nail it down more specifically than this is beyond my expertise -- e.g. the above two course descriptions were intended to convey this distinction but looking back it seems even more complicated: none of EMAT 3800 would be appropriate material for aspiring teachers of most subjects other than mathematics, and some but not all of EMAT 3900 would.
The two linked wikipedia articles give entry points into the vast body of literature
on these matters; people who were interested enough to read this far are encouraged to delve into the literature itself. And when you do, come back and tell me about it! I am an academic mathematician and thus a mathematics educator, but I have no specific training in mathematics education.