I'm an "old-ish millennial" (born in '86), so I see both sides of this, actually. Like Texstyle says, J Crew, along with Banana Republic, sort of taught me "how to dress". I didn't always dress nicely, of course, being a rebellious youth, but when I did, that was my template. Still is. Well-fitting trousers, an interesting shirt, a menswear-inspired jacket when necessary.
When I started really re-building my wardrobe three years ago (after a couple years of travel and then having a baby), I felt completely betrayed by J Crew (and BR and Gap, for that matter, and Land's End and LL Bean and Eddie Bauer -- all the names that were synonymous with quality for my parents' generation and when I was a kid). The quality was gone, everything fit oddly, and some of it felt like it could have come from Target or Old Navy. It took me quite a while to find my footing in the new retail landscape, and come to terms with the fact that the brands I'd considered old standbys just weren't anymore.
My longest-standing articles of clothing are J Crew, from the early '00s. They're getting a little rough, but they still ooze a level of quality and character I don't really see at that price point anymore, especially from big-name retailers. Taylor Stitch is doing a good job of carrying that torch, IMO, in terms of quality, fit, design, and even the cohesive lookbook style storytelling. I've ordered two things from them so far and both are slam dunks. I'll definitely be back for more in the fall. I wish I could say the same of Everlane, but I find their quality just a bit too flaky and their sizing/fit is hit-or-miss. And my basics mostly come from Uniqlo -- they've more or less replaced Gap for me -- solid-colored, mostly- or all-cotton at a reasonable price, nice selection of colors and graphics, etc. So yes, the fall of the retail giants has pushed me towards newer "millennial" brands. And the simplicity and anonymity does, admittedly, appeal to me. People my age (or at least me) grew up under the influence of Naomi Klein's No Logo and Adbusters so maybe that underpins the aesthetic to some degree -- but then, we're all wearing Adidas again, so maybe we haven't come as far as we think, or escaped the siren call of kitsch and irony entirely...