Repeating here so people can see, since this post is alarmingly misleading. This logic is so common it is a recognized fallacy in the field, immortalizing a Harvard man who tried this for political reasons. That being Richard Lewontin. What he says isn't wrong (unless you look at the whole genome), but it does not invalidate the concept of race at all since the odds of two people from different races being more alike than two people from the same race is 0 with enough loci being used (obviously) https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1893020/
 
Ctrl-F 'thousands' to find the relevant part, not the abstract.
 
Your post from the american anthro society is repeating something literally called "lewontin's fallacy" and all the other justifications are nakedly based on their 'values' rather than an objective criteria. The idea of race is that there is this additional dimension of variation on top of ordinary individual variation. Still recognizable humans, but clearly different phenotypes based on ancestry and ancestral environment which are inheritable and predictive, with correlations to other racial traits.
 
Your post about dog breeds uses arbitrary criteria and thresholds which have no inherent meaning really. All that selective breeding did for instance is making the differences occur more quickly than they would in nature (there are naturally occuring dogs and cat breeds after all). The poster is right, this logic could be used to destroy all kinds of categories in biology, maybe species would be spared but even then there are enough edge cases (like lions and tigers actually producing fertile offspring, or human-neaderthal offspring) that someone could creatively deny it as well.
 
There's no 'threshold' of heterozygosity though like you seem to imply. This is just arbitary, and usually motivated. What matters for the concept of race to be sound is if it is identifiable, inheritable, and predictive. It is all of those things, even Dawkins goes into its taxonomic significance.
 
There's about as much genetic diversity among human races as among races of chimpanzee
 
https://www.ncbi.nlm.nih.gov/pubmed/19695787
 
Hell, there are species with far less heterozygosity which have recognized 'subspecies' which is more than mere racial classification.
 
And also, the literature in the field uses race still.
 
There's no way you can really justify your position scientifically, you either use outright fallacious and motivated reasoning (from anthropology) or arbitrary criteria with the comparisons to dog breeds
 
http://blogs.discovermagazine.com/gnxp/2012/05/richard-dawkins-accepts-the-usefulness-of-race/#.Wnt-I-jwaUl
http://thealternativehypothesis.org/index.php/2016/04/15/329/