I'm still asking this question, and in fact I'm currently exploring different ways these processes — usually performed un- or sub-consciously — affect our disciplines in dramatic ways. In my reading on "parting of the ways" scholarship, I've come across a mass of confusion regarding precisely this question [again, How did we come to identify the NT texts as what they are?], but interestingly I haven't yet found anybody asking this question explicitly.
In 2006 James D. G. Dunn's book, The Partings of the Ways [orig. published, 1991], was published in a second edition. Significantly, the second edition has a new preface and an essay exploring the relationship between the Mosaic and the "new" covenant(s) (cf. Jer. 31.31–34); these later additions call into question most of the conclusions from the original edition, but the substance of the book hasn't (as far as I can tell) been edited to address these new questions.
There are a number of things both commendable and questionable about this book (as with all of Dunn's work, in my experience). But I want to call out a particular formulation by which Dunn introduces his discussion of Matthew's gospel. Dunn starts,
It is worth giving particular attention to Matthew, since his is the most Jewish of the Gospels, and since his christological claims mark important developments or phases or features in second generation Christianity. (2006:278)
Throughout the rest of the book he spends much more attention, among the gospels, on Matthew and John; Mark and Luke, by comparison, are marginal. And while his treatments of both Matthew and John make a number of points worth considering and following up, I can't wonder what it means to label Matthew as "the most Jewish of the Gospels," or why Dunn would want to use such a label.
Dunn's statement directly raises the questions I posed above. What makes a text (or an individual, a group, a practice, a belief, etc.) "Jewish," and what distances a text (and so on) from that label? Why is Matthew characterized as "most Jewish" while the other three are somehow "less" or even "non-Jewish" (though, to be fair, I suspect Dunn would object to this latter formulation)? More fundamentally, are these judgments at which we have arrived on the basis of textual analysis, or are these presuppositions that will determine textual analysis? While this last question is certainly posed in terms too stark to be helpful, I would like to see NT scholarship pay more attention to this latter possibility.
I suspect the "Jewishness" of the NT texts and the events and figures portrayed therein will become a prominent theme on this blog. Hence the all-caps in the label attached to this post: I use JUDEO-Christianity in a way similar to the identification of genus and species in zoology. You only need read one or two chapters of Partings to see how fundamentally different this conceptualization is from Dunn's, who writes of "four pillars" of Judaism and asks throughout the book whether NT formulations and ideas call one or more of these pillars into question.
One last point: My own use of JUDEO-Christianity, like Dunn's use of "Judaism" and "Christianity," similarly risks reifying both as "things" that exist rather than spaces that are contested and fought over discursively (and, all too often, really). But hopefully thinking of "Christianity" as a species of "Judaism" does better justice to the situation in the first century/-ies in which, unlike our own situation, "Christianity" and "Judaism" were not separated religious identities. But more on this later.
UPDATE: In the CSB Harmony of the Gospels, which I am using for my first-year gospels course, Rick Melick goes even further than did Dunn: "Matthew is at once the most Jewish and the most universal of the Gospels" (2007:11). I am amazed and disappointed that scholarship this recent can make such claims, not just about Matthew being, again, "the most Jewish," but also that it can imply that Matthew's double-status as "most Jewish" as well as "most universal" stand in tension to one another. Perhaps I'm over-interpreting here, but the assumption behind this phrase seems to be the old "the problem with Judaism is that it's too particularist [nationalist?])" prejudice, combined with the inappropriate view that what was right about Christianity was its universalist, non-nationalist emphasis. Such blunt and unwieldy language used to analyze Christian origins and Christian texts strikes me as similar to trying to carve Michelangelo's Pietà with a backhoe.