Not picking up on the correct species is definitely an issue. Misidentifying a large dog as a chihuahua is also a problem. I'm inclined to blame that one on human error, with the wrong sample getting reported. Sloppy sloppy.
As far as the Great Dane coming up as 1/4 Staffordshire, that's not ideal, but I can see how it might happen. Great Danes and Staffies are both descended from the same ancestral stock: mastiffs used as catch dogs for boar hunting and as guard dogs in the 1500s. I've heard of other purebred dogs coming back as a cross between two closely related breeds, e.g. Standard Poodles coming back with a bit of Barbet listed in their ancestry. I'd be much more concerned if the test reported a completely unrelated breed, such as Akita or Irish Water Spaniel. I notice the two most established companies, Embark and Wisdom, didn't make the erroneous report; they have been around long enough and have the financial resources to have tested a wider range of purebred dogs.
As far as the confusion over the two village dogs, that's really just Eurocentric chauvinism. A breed implies the dog is the product of a controlled breeding program. Most "purebred" dogs, the way we think of them, have ancestors recorded in one of the big five western breed registries. These registries are all relatively recent. The KC was founded in the 1870s, the AKC and CKC in the 1880s, the UKC in the 1890s, and the FCI in 1911. So, for practical purposes, any dog that we would consider part of a breed has ancestors or cousins that came to the attention of a breed club in either North America or Western Europe some time between the end of the Victorian era and today. That includes non-European breeds like Basenjis, Canaan Dogs, Shih Tzus, Akitas, etc. Some westerner got hold of specimens of these dogs, established they could breed true, and lobbied a kennel club for recognition of the breed.
So, what about all the places in the world that aren't North America, France, or Germany? Well, the locals absolutely weren't waiting for a westerner to show up like Prometheus giving the gift of dogs to a grateful local populace. They had dogs already. They've had dogs for thousands of years. They just didn't have dogs recorded in a western stud book. Why would they? Therefore, it's totally unsurprising that a dog from those regions might test as something that doesn't really match up neatly to a western breed. Why would it?