quedasdilla
Yes I see your point 
My problem with the word "heritage" in this usage is that it suggests a deep cultural connection
I don't think thats true really, I have scottish heritage, but I don't have a particularly deep cultural connection with scotland, but I still very much acknowledge it.
In this case it just isn't true that there is a "heritage" in the proper sense of the word
Well, obviously from my point above, I would disagree with that, heritage isn't something you can deny really IMO.
It assumes that "identity" is an automatic product of your racial background and I just don't think it's a simple calculation
I sort of agree, as I don't think in all cases identity is automatic, thats sort of my point.
A child can grow up from an ethnically pure background (black, white or whatever) and identify completely outside that background for reasons known to him or herself
Yes maybe, but they would still have the same heritage regardless.