"Black at heart" used to mean bad, and have nothing to do with skin colour. Do you refer to a new meaning, ie that someone can feel like a black person in a white body, kind of thing?
If so, then I think that claiming to be "Indian at heart" or "Scottish at heart" is slightly different to be claiming to be "black at heart". Because the former could mean that you feel a deep connection to the culture, or the landscape, or the language of a particular country. (But could still be rather iffy, if you are not Indian or Scottish, and you define "Indian" or "Scottish" as a set of dodgy stereotypes).
The latter, identifying as a skin colour "at heart" seems worse to me, as it suggests that everyone with a particular skin colour is united by some "feeling", when in fact you can't reduce everyone in the world with that skin colour to a particular culture or "feeling".
I don't know if I'm explaining myself very well. But I would be pretty sceptical of a white person claiming to be "black at heart" in that context. I would assume, at best, a severe lack of self awareness and an abundance of self importance.