Interesting question. I've seen it before, as well, but never thought about it like this.
A starter question, I think, would really be what the exact Greek/Latin terms the various Fathers used for the words translated as "Christian race" by, presumably, 18th/19th century pedagogues who had an almost obsessive desire for taxonomy and classification. Do you have any cites for the phrase that our more erudite posters who have Greek and Latin (and access to the citations in the original language) can give us an idea of the original wording?
Secondly, living in an age where Rome (and later Constantinople) was supreme and the Great Uniter, so to speak, I think that the Ancients had a much different view of what constitutes ethnicity, and its importance. It was far more important to be a citizen of the Empire than anything else, regardless of ethnicity. It was, after all, what saved St. Paul from crucifixion even though he was a Jew.