It sounds to me that the midwife dealt with the issue very badly, and offended you in the process.
DD is white, and she had the jab at two days old as DH is from another country where TB rates are high. We don't often go there, or have visitors, but the vaccine is quite low risk so we saw it as a good thing to have done.
The logic behind offering it to your child but not other children is sound, but woefully explained. It is not about skin colour, but the possibility of increased exposure to TB.
I do feel quite strongly that this should be explained to women whilst they are pregnant. There was no mention of it to me until DD was two days old and they turned up with the needle and vaccine. There wasn't really the time to think about, research and understand it, and given that some (many?) women are exhausted and emotional just after having a baby, it is possibly not the best time to spring this on them!
Perhaps in the maternity records, on one of the "tick box" lists in preparing for the birth and developing the birth plan, this could be included?