It's very clear that language use is occurring. A recent video shows the owner on the phone with the vet, and when she says "come in to express the anal glands" Bunny immediately gets up and uses the "no" button.
Another dog, while its owner is blowdrying her hair, presses the "wet" and then "dry" buttons.
Otter, Bunny's brother dog, learns the context of words from her, and they use words to each other, with clear, contextual meaning. They express humor, sadness, anger, frustration, empathy, caring, happiness, apathy, excitement, and more.
Their grammar is limited, the processing they can do is slower and on a smaller scope than humans. They definitely lack the breadth and depth of human cognition, but I can't understand how, given the overwhelming evidence of deliberate use of language in complex, nuanced, abstract, emotional, contextually relevant ways, people insist that "well that's not what they're doing!"
I think it's very likely that any and every mammal with a brain above a certain size will be able to use language, given the appropriate tools for it.
People are quick to point out the story of Clever Hans, but I think that story is worth revisiting. There's a very powerful bias for people to hold humans above other animals as somehow intrinsically special, fundamentally different from all other creatures, and language use being somehow unique to humanity seems to be one of the most stubbornly held beliefs.
I think we have language because we have vocal cords, complex mouths, opposable thumbs, fingers, and very large primate brains with a proportionally massive neocortex.
Take away the hands and we lose tool use, and probably can't develop culture, and so never develop language or complex vocalization, and never garner the benefits of those things. Take away culture and you have humans living in feral conditions. Modern studies of language deprivation, children raised in feral conditions, and other situations show us that it looks like some humans lose the ability to learn language past a certain age under those conditions.
Take away the effective mouth and vocal cords of human biology, and we may never have developed spoken language, but would likely have developed signing and nonvocal audible communication methods, and then developed culture around that.
So knowing that, when you take another look at the talking dogs with buttons, it's worth considering that up until a few years ago, people had essentially raised their pets in the absence of culture. No efforts were made to teach them language in the context they'd be able to handle. They didn't have tools that served as vocalization, limiting their effective vocabulary to bark, howl, sneeze, and whimper. Some dogs, through care and exposure, were seen as exceptional if they picked up words through context and repetition, like toy names and so forth.
If a standardized vocabulary was made available, with a repeatable training framework for dogs and cats, we give them what amounts to a culture prosthetic, and buttons give them a replacement for mouths and vocal cords.
It shouldn't be unreasonable or even particularly shocking to consider animals with brains similar to our own being capable of language use.
Imagine the conversations you could have with an orca trained to use buttons, or a pig, cow, bear, lion, or whatever your favorite mammal is.
I think we need to be much more open minded, not overly skeptical, and stop trying to find ways of insisting on human exceptionalism. It might help us learn more about how language and cognition work, and what it is about human brains that gives us such an apparent edge. Or maybe that edge isn't as significant as we think?