All the comments so far seem to forget that blind review goes in two directions. A senior faculty member in my department likes to point out that she very often knows who wrote the articles she reviews, but the important thing is that the authors don't know she's the one reviewing them -- which means she can be fully honest in her assessments of their work.
That said, given what we know about unconscious bias, it is important for reviewers at least to be self-reflective about this. Which I understand some are not very inclined to do.
It seems to me that this faculty member has it all wrong. (I'm not sure exactly how much of this the anonymous commenter believes, and how much s/he is merely attributing to the faculty member.) Blind review goes in two directions, and this means that the referee is not supposed to know whose paper it is. Because, as the anonymous commenter notes, there are loads of unconscious biases, and blind review is supposed to control for them. But it is not enough to be "self-reflective" about this. If the biases are unconscious, it is literally not possible to correct them via self-reflection. The way to correct them is to eliminate the bits of knowledge they operate on. And so the effective way to be self-reflective about latent biases is to acknowledge that they are there, and to realize that blind review procedures are the only way to protect against them, and to observe those procedures.
Or am I missing something?
P.S. The faculty member is right about how the author shouldn't know who the referee is.