This obstacle is no different than that for transgenders whose claims that they "feel" like a woman are meaningless unless they presume to know how men and women feel differently from each other. Likewise, our efforts to honor and respect those transgender claims is an endorsement of those assumptions of psychological sex differences.
No, on two counts. First, all that needs to be acknowledged is the experience of dysphoria. No individual can truly claim to know what it feels like to be a man or a woman beyond their own personal experience, but they can know when their mental experience and their anatomy cause a feeling of discrepancy.
The same consideration may or may not apply to this transracial scenario. If so, it's the feeling of dysphoria we have to consider. I have no opinion on whether this transracial experience is at all similar to gender dysphoria, but I would say that this woman does not need to know what it feels like to be black by some (as of yet undefined) objective standard to feel she identifies more with looking black than white.
Two, science has some rough understanding of differences between male and female neurology and the impact of hormones. While not every individual transgender person is going to be tested, some research has been done indicating neurological differences between trans men and cis women, and trans women and cis men. It's imperfect, like most science, but there is some data reducing the amount of assumption required. I am not aware that science has examined racial identity the same way. If we compare being transracial with being transgender, our understanding of the latter has advanced a lot more than the former. Maybe being transracial will follow a similar path, or maybe it won't (or cannot). Time will tell, but at present time the two scenarios are at very different places in our level of understanding and... assumption.