I can see if I can find this for you, but I believe it was a Russian biologist who decided to test this theory by breeding Siberian wolves. He took a few, and allowed only the most docile males and females to breed. He occasionally brought in fresh wolves to replenish the gene pool. He bred them, if I recall correctly, for 20 generations.
At the end what he had were still genetically Siberian wolves, but they were much smaller, with floppy ears, spots and acted domesticated. If you didn’t know that they were wolves, you’d think they were dogs. And what he found of interest is that these traits arose from breeding them toward being docile.
So it’s not just DNA. Stuff happens in the interpretation of DNA that makes everything a bit less clear.