I agree that the left has its own form of bigotry that goes underchallenged for the most part, but the argument you're referring to is one of the most insidious and disingenuous the right espouses.
For the most part, the identity politics of the right have evolved from an explicit racism and sexism to an implicit one. The argument goes "as long as we don't explicitly mention race, sex, or ethnicity, we're ok," which isn't true. Voter ID laws are perfect example of this: they deny citizens of their constitutional right to vote without due process, in the absence of evidence for any systematic problems, and disenfranchise people based on socioeconomic status, and implicitly by race, but then complain when they're called on it, whining that the left has somehow shifted the goalposts. It's a sort of Bart Simpson defense ("I'm just walking around swinging my arms and if anyone happens to be in my way, it's their fault").
The essence of the Trump phenomenon to me seems to be the right getting tired of playing this game, of hiding the racism and identity politics they play. Trump just came out with it unapologetically, "saying it like it is" according to some of his supporters. I'm not saying everyone who voted for Trump did it for those reasons, but it was certainly a big part of it, especially during the primaries. I believe much of the mainstream right was not truly appalled by what he said, just the fact that he was so open about it. They distanced themselves from him when they believed it was politically harmful, and now are cozying up to him again when they realized it wasn't.
Who's playing identity politics again?