The placentum provides surface area for diffusion between the distinct bloodstreams, yes. I believe it's possible for Mom and kid to have incompatible blood types. The author is making a rather far stretch from this data point, however.
I don't think that blood cells normally cross, but Rh factor incompatibility often causes problems. It mom is Rh negative and the child is Rh positive, the mother may develop antibodies that can harm future Rh positive fetuses. Generally, however, this mixing of blood happens at childbirth and the antibody reaction is preventable by drugs given at childbirth.
Do you mean her point about the Bible making reference to genetics before genetics were understood by man? If so, how?
In my opinion, even if she erred on the point of blood intermingling, that does no harm to her argument about the Bible and science.