It's a fair question. And I think that logically it's equivalent.

However, to maybe see why people don't do it (I imagine, I have no personal experience on this), let's put it in a US perspective. What if a visibly black person identifies as white? Or vice versa? Do you think it would be taken seriously by even the most ardent liberals?

EDIT: For context, let's remember the case of Rachel Dolezal [1]

[1] https://en.wikipedia.org/wiki/Rachel_Dolezal