Ah, but, "Walk around like you own the place" is The Doctor's M.O. That is the way he always approaches every situation and often works (and often doesn't, sure, but, that is his typical attitude and advice). He didn't tell her that because she's black and was concerned about prejudice, he told her that, because that's his typical response.
I'd say it's especially in character for 10. I've been rewatching some of S3 and we can see he's similarly unconcerned about what Martha has to sacrifice in Human Nature/Family of Blood or about the feelings she has for him. And he seems to me dangerous to be around, I guess the Doctor always is but 10 seems quite ready to give a "sorry, so sorry" without missing a beat.