He's not old-fashioned, exactly. He's just wrong. The problem is that people have a tendency to mix up biological sex with social gender (not to mention overlooking the various genetic variations in the sex chromosomes, such that woman and XX don't actually overlap perfectly anyway).
In terms of his talk, why is the idea that "the self ...may be plucked from the shelves of a ... supermarket" so outrageous -- as opposed to the traditional approach where, it seems, the self is often what society tells you it has to be? When it comes to sex and gender, this idea is at least grounded in some amount of logic, although it's rather too dogmatically enforced at times and there is more to it than that even biologically. In many other cases, it extends even further than that (castes in India, etc), and people's entire life path is mapped out for them with minimal input from themselves.
What we're seeing these days, then, is a reaction against the older approach and a transition to a society where personal identity is far more up to the person than it is to anyone else. As with any transition, it's invariably going to be messy, with some people being "left behind", and others perhaps pushing the matter too far. At some point, things will settle down into some kind of balance, where people are who they choose to be and, more importantly, nobody actually cares.
At any rate, no matter your personal perceptions of who someone "actually" is, when it comes to their gender, it seems a matter of basic courtesy to respect their 'choice' of identity, even if you don't agree with it.