I believe, and anyone feel free to correct me, that he's considered the self-insert character because he is the character we want to be in this universe. We want to bang Matt Murdock. We want Matt Murdock to be secretly in love with us.

I also believe, once again a correction may be needed, that he's also considered the self-insert character because in the show there is a canon romantic relationship of sorts with Matt - it's Claire. And there isn't really a whole lot of Matt/Claire out there. Hell, it could even be Karen but once again, not seeing a whole lot of Matt/Karen. Why, when there are two obvious romantic pairings given to us in the show, do we fall to the unlikely pairing of Matt/Foggy? (Because, let's face it guys, I really doubt Matt/Foggy is going to become canon.)

Now, these are just my observations on the matter from reading a whole shit ton of fic and seeing a whole lot on tumblr. I hope somebody with better insight comments on it.