The phrase "false myth" is ambiguous because "false" can mean both "untrue" and "not genuine/real".
By this definition, then, a "false myth" is something that it is not a "true myth", that is, it denies the definition of a "myth". As I see it, it can either be that it is not a "myth" because it is not widely believed, or because it is not false.
This meaning makes sense, but it is probably rarely used.
I've seen a lot of times the phrase "false myth" used to emphasize the falseness of the myth.
This makes slightly less sense. I don't like it. Because "myth" implies false, my initial interpretation is to assume that "false" in front of it must mean something different.
Because of the ambiguity, the phrase appears to be an auto-antonym, a word (or in this case, a phrase) that is its own opposite, and can only be understood by context. It should be avoided. Instead of "X is a false myth", say what you mean:
X is popularly believed to be a myth, but it is not.
Or for the emphasis meaning:
X is a ridiculous/ludicrous/wild myth.
For the first meaning, you can also use "faux myth". The word "faux" can only mean "not real", so is not ambiguous.