That's definitely not what the left asserts about identity. That's what the right wants you to believe the left asserts about identity.

The left is merely pointing out our society has been deliberately built in ways to benefit certain groups of people, and that if we truly believe in equality of opportunity we should remove those structures. If you look at the history of this country, that point is painfully obvious.

The right on the other hand is committed to preserving those structures that unbalance the playing field and gleefully pretends that they don't exist and were never built.