Who's asking?
Lawyers have a simple way of dealing with "What is a ...." questions. We ask "For purposes of what decision?" Why would I want to know what a concept is? What decision that anyone is going to make depends on whether X is a concept? If you can't give an example of "If X is a concept, do A, but if X isn't a concept, do Y," why would you even ask. And even then, wouldn't the question be "Is X a concept?", a question that one might be able to answer without defining the limits of the term? Can you tell me whether Bermuda in the Atlantic Ocean without telling me precisely where it begins and ends?
Otherwise, aren't we in Lewis Carroll territory? When Frege uses the term, doesn't it mean what he intends it to mean?
My approach to these subjects is to try to get rid of the nouns and adjectives and deal in verbs. What does the word "concept" do when we discuss? Thus, to use a concept, the concept need not "exist" as a concept. Rather, the user needs to be able to conceptualize, and if the user can conceptualize, the user is able to communicate efficiently.
I would offer the analogy of using Cartesian coordinates to communicate information about a line. Anyone who can conceptualize can learn that a straight line can be represented by the equation y=mx+b. So, I can say to someone who can so conceptualize and has so learned "Line, 4, -1" and that person can see a steeply ascending line that crosses the Y intercept just below the origin. I have communicated using the shared "concept" of representing a line as an equation yielding Cartesian coordinates.
In a communication about lines, there is a part that is always the same - the form of the straight-line equation - and a part that is always different - m and b. By using concepts, we can focus on the m's and b's, because the concept takes care of the rest. A concept thus does what a template does: it converts an explanation into a fill-in-the-blank exercise. But the "concept" isn't actually doing anything. Rather, the discussants are conceptualizing.
Is there a reason to reify this "doing"? Sure, but only so that we can shorthand the verb version. There is no need, I submit, to explore the nature of a "concept," because it is already just short-hand for how discussants get on with their business. Why ask more?
In one of his books, Richard Feynman talks about someone not grasping how quickly he, Feynman, did "information transfer." That's because Feynman thought in concepts. He could quickly see what part of what he was learning about was just like (used the same concepts as) something he already knew. So, by all means, give this thing a name. But I'm not persuaded there's much reason to dig any deeper into it.