A bit misleading in the title of the video. The question posed is whether they have the souls of men or the souls of beasts, not whether they have souls at all. If that were the question, it would be rather simple. Do men have souls? Do dogs have souls? If the answer to both is yes, then it would stand to argue that they do.
Also, I think he's mispronouncing "Cynocephaly", but it could be correct for the time. I don't know enough about medieval pronunciation.
The question is posed as a matter of the nature of their souls, which is not surprising, being that this is correspondence between religious scholars. It's also trying to fit them in the two known possible categories for life on Earth, according to their doctrine, again, because they are religious scholars. He makes a decent analysis considering the framework that he is bound by.
But I think this framework does inhibit the understanding of the possibilities here. Whether or not we want to get metaphysical and call it a matter of the nature of the Cynocephaly's souls, the core issue remains the same - a discussion about personhood. Are the Cynocephaly people, animals, or something in between? That third option is one that they don't discuss in the letter, because it's not really part of their framework, and not part of our usual understanding.
We certainly grade animals on intelligence, but we still generally have a hard distinction between man and beast. There is quite a gap between human capability and accomplishments, and that of any animal. The word 'animal' itself, when not used in a biological context, establishes this dichotomy, as it specifically excludes humans from other members of the animal kingdom.
This is a constructed dichotomy, based on our perspective. And for most purposes, it works. There aren't really other animals that we acknowledge as approaching human intelligence, but falling short. At least not in any meaningful way. But this is because of a mix of the random chance of evolution, and the fact that our ancestors probably murdered or interbred with any subspecies of human that could fit into that description. None of this means that such a situation can't occur, though.
Indeed, there are some people that are challenging this dichotomy. And not all of them are insane assholes from PETA. Case in point, the
Great Ape Personhood Movement. The argument that the level of intelligence in other great apes should afford them certain basic rights, such as disallowing keeping them in captivity or using them for scientific testing. There are similar arguments for other animals, like cetaceans, octopi, or
fucking trunkies elephants. The issue of whether or not this is actually a good idea, or will snowball into absurd insanity, notwithstanding, this does show some of the possibility of there being an 'in-between' in the man and beast dichotomy. Obviously, you can't recognize a chimpanzee as an equal to a man, and expect them to function in a human society. You can't hold them responsible for understanding and obeying laws, and thus you can't afford them full human rights. They are fundamentally lesser beings, but that doesn't mean we shouldn't have standards for how we treat them.
A more poignant example of this problem will probably show up sooner or later, whether it's in the form of space aliens, AI, or if Elon Musk finally engineers catgirls. We're going to have to make a call for how we approach another intelligent life form, and we may not be able to establish them as either sapient equals or animals. Even a lot of science fiction makes this mistake, where all aliens are either animalistic, or of roughly human intelligence, like there's a hard line between them. Even if they are of similar intelligence, that's still no guarantee that nature will be compatible with that of humans in such a way that we can treat them the way that we treat our fellow man.
