So, even if came back with some crazy animal mixture, it could be true then?
I'm not sure what you mean here.
For the example I used with the Arabian horses, what they do is match individual markers in the DNA by comparing the parent DNA with the offspring. DNA is like a fingerprint, unique to each individual, and some things are passed on from parents, so you can compare the samples and get a reasonable idea of parentage (obviously a huge oversimplification). So it doesn't tell you anything about what breed the animal is, just that this one horse is the offspring of these two particular horses. So since those two horses are purebred Arabians, so is the offspring. If one or both of the parents didn't match, it wouldn't tell you anything about who the actual parents were
, just who they were not
. The horse could be a purebred from two other Arabians, or it could be a Clydesdale for all the DNA test would tell you.
With DNA testing for genetic diseases, again, they're just looking at a specific part of the DNA. So let's take a common problem in GSDs, hip dysplasia.* They would take a DNA sample from a dog and look at the particular genetic markers they have identified as those that affect dysplasia. From this, they will be able to determine the likelihood of the dog passing it onto its offspring. However, it still doesn't tell you anything about what breed the dog is--hip dysplasia is found in all sorts of different breeds and the presence or absence of it doesn't tell you anything.
I guess a general way to explain it is that DNA testing (in this sense) is all about comparing samples to other samples. The problem with breed tests is that many of the markers they use aren't actually accurate (as in, they're shared by many breeds), and can vary between lines within a breed, etc. So while there's always a possibility
that the test is accurate, in reality you're just as likely (and possibly more so) to be accurate by making a visual assessment and guessing based on that.
*I don't actually know that much about the state of genetic testing in dog breeds as I don't breed and have rescued all my dogs so never had to learn, but I didn't want to use another horse example.
The process is the same, I just don't know exactly what conditions have a test.