Incidentally, you don't hear people saying that their dog behaves like a cat. Interesting that. I think the cat-behaving-like-a-dog cult is part of the internet fad about cats; usually funny pictures of cats. Without wishing to come over as a grumpy old man, I think it is slightly disrespectful of the cat.
There is a lot of overlapping behavior in cats and dogs. This is particularly the case with respect to substitutes for hunting prey. I am thinking of playing fetch. Cats will play fetch. Many cats play fetch. This is not dog-like behavior. It is simply the behavior of a domesticated predator expressing deep seated behavioral drives. Both dogs and cats are predators.
Examples of other forms of cat behavior that people think are dog-like are:
- wagging tail
- sitting on command
Panting is not uncommon in cats. Cats don't cool themselves by sweating. They cool themselves by evaporating saliva on their fur and some sweating on the feet. But panting is also a way of cooling down. Stress and illness can cause panting too.
The cats in videos who are panting are also playing fetch so they are hot. The two go together naturally. This is not a cat who thinks she is a dog or cat behaving like a dog. She is a cat behaving like a cat.
A cat can be trained to do all sorts of things one of which is sitting on command. This a person training a cat to do something that dogs do. That is all it is.