Also an experienced meditator.
Not everyone thinks in words. Some people think in pictures. Or in other sense consciousnesses. As an experienced cook, I can think in taste and smell without any auditory component of 'what should I put in this dish'. For example.
As for word thinking, passive thoughts are more auditory and active thoughts more somatic (throat and jaw muscles will move). These can be decoupled from the sense of I making, especially passive thoughts.
At which point you get thoughts think themselves, to quote Jack Kornfield. A sort of bubbling up of passive thoughts in voices that aren't mine.
It's likely animals that live close to people experience this. The owners voice yelling no when they do something the owner wouldn't like, even if the owner isn't around.
Anyway, trying to not think can be like holding your breath. I can do that for awhile. But it's not right effort. Letting thoughts settle, like sand in a glass of water. And letting go of the sense of I making. The mind will rest quite naturally. That's calm abiding.
In other words, it's attachment to the inner voice that's making it difficult for you to imagine that a lot of people think in pictures or other ways. And noticing this sense of attachment in your practice with the intention of letting it go, might deepen your insight into yourself and what others may or may not experience.
Edit: this listening meditation is helpful for me in letting go of attachment to the inner voice. As is annapanasati, especially the third tetrad.
I'm sure it depends on the animal. In fact humans think in at least two inner voices.
The ear consciousness is receptive, and the speech consciousness is active.
What I mean by this is a dog or another animal that spends a lot of time with people likely has a passive inner voice of their owner. A dog might hear an owner yell no when they go to do something the owner doesn't like, even if their owner isn't around.
Animals that are capable of speech such as a parrot, will likely have an active speech consciousness. Which is more somatic in tone.
For example, when I am in active speech consciousness I can feel my jaw and tongue muscles move. When in passive listening consciousness, my ears might move or strain to try to hear the inner speech.
With practice these somatic sensations can be decoupled from their internal sense consciousnesses. Which tends to help them quiet down and deepen meditation.
This is one of my favorite practices that's accessible for people who don't really meditate. The guided meditation is the first fifteen minutes of the video, so you don't have to listen for the whole hour to get an inkling of what I'm pointing at.
https://www.youtube.com/watch?v=7OW9LNSVjPo