Does it matter?

The real answer is it depends on what box they check when they fill out forms. Presuming they don't check the "choose not to answer" box. Then there's simply no way of knowing.

Now that is a complicated question. There is a "one drop" narrative that extreme supremacists have... but someone who doesn't look black claiming blackness would definitely get a side eye from a chunk of the black community where even lighter skinned black folks are made aware of the privileges of shade and tone...

So probably both yes and no or the principle of Mu depending on circumstance?