I'm trying, but I don't see it at all with these examples.
1) just seemed like random pressing until the dog pressed "paw", then the owner repeated loudly "something in your paw?" The dog presented its paw, then the owner decided "hurt" "stranger" "paw" was some sort of splinter she found there. The dog wasn't even limping.
2) I didn't get any sense of the presses relating to anything the dog was doing, and since the owner was repeating loudly the thing she wanted the dog to find, I was a bit surprised. Then the dog presses "sound," the owner connects this with a sound I can't hear, then they go outside to look for something I can't see.
Billie the Cat: I simply saw no connection between the button presses and anything the cat did. The cat pressed "outside" but didn't want to go outside. The cat presses "ouch noise" and the owner asks if a sound I didn't hear hurt her ears. Then the cat presses "pets" and the owner asks if the cat wants a pet? The cat presses "noise" and the owner continues the monologue apologizing for the painful noise and offering to buy her cat a pet. Sorry to recount most of the thing, but I don't get it at all.
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Not trying to debunk talking pets, but I'm not seeing anything here. I at least expected the dog to be smart enough to press particular buttons for particular things, but I suspect the buttons are too close together for it to reliably distinguish them from each other. I'd be pretty easy to convince that you could teach a dog to press a button to go outside, a different button when they wanted a treat, and a different button when they wanted their belly rubbed. In fact I'd be tough to convince that you couldn't teach a dog to do that. Whatever's being claimed here, however, I'm not seeing.
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edit: to add a little more, I'm not even sure that *I* could reliably do what they're claiming the dog is doing. To remember which button is which without being able to read is like touch typing, but worse because the buttons seem to be mounted on re-arrangeable puzzle pieces. Maybe I could associate the color of those pieces with words, but that would only cover the center button on each piece.
If a dog were using specific buttons for language (or if I were doing the same thing) I'd expect the dog to press a lot of buttons, until he heard the sound he was looking for, then to press that button over and over. Not to just walk straight to a button and press.
I think the cat just presses the buttons when it wants the owner to come, and presses them again when the owner says something high pitched at the end and looks at it in expectation.
Consider this older one where there weren't as many words - https://youtu.be/6MMGmRVal6M and https://youtu.be/FPC6ElzSdxM
Or for billie the cat - https://youtu.be/DiuQqgTw-jY
The point with these is that there is language related thought going on there and an attempt to communicate from the dog (or cat) to the human.