*armchair animal psychologist cracks knuckles*
Way I see it, those dogs are very nervous, rather than aggressive. It's the reverse of the normal cat meets dog at home scenario. Usually, the dog is bigger, confident, and will go and see the cat. The cat will swipe and hiss to keep the dog at a distance until it feels comfortable. But here the cat looks to be gently trying to get close.
Also, I'm going to stick up for dogs like these as estimable foes, especially if they are at all Daschunds - once bred to kill badgers, even used in hunting deer. Dogs are pack hunters, so if they were 'attacking' like the video says, I reckon these little blighters would round it up between the two of them, keeping low, keeping it spinning round, going for bites of easy to reach areas, looking to get actual bite wounds to try to weaken or immobilise it, rather than get risk getting caught in a corner as they are here.
Now the first dog is really, really anxious - licking its lips because the adrenaline dries them out... is it really fair to call it aggressive? It's definitely snapping and barking as close up to the face as it can to warn the cat how quick and fierce it is, without daring risk an actual nip. Makes the biggest impression. Sort of like a cat might swipe and actually let the dog feel it's claws, without taking it's eyeballs out. Personal space, dude.
I reckon this squabbling descends into less and less fierce snapping and nibbling until they are soft mouthing and licking each other.
I've got this ...uhhh.... 'pet' theory that most dogs that 'play' upon meeting are doing something similair but less fraught... all the running around seems to be them determing who has the mobility vs agility, strength/stature and reach, who will stalk, who will chase, always who can get a cheeky little grab or nip in. Who is aggressive, who is cool and confident, who stops you dead in your tracks if you are about to end up face to face with them. Even the bark tells the other dog something, implying chest depth and size of frame. All works out in silence, in case they should happen to meet a predator or prey. You can see in dog packs when they live and hunt together they are all about who has which role.
Probably bollocks but I love talking about it!