In the way of responding to challenges, you can't really distinguish between cat and dog because they're fundamentally the same, and depend on temperament and experience. Some cats are skittish, some dogs cower. Some cats hiss, some dogs bark. Some cats bring out the claws, some dogs bite.
You´re kidding? It´s completely different on average. Cats will generally not get into a challenge unless they feel the "other side" is specifically a competitor for itself, and >9 times out of 10, those times will end up as staring contests.
Challenge a dog and most of the time it will challenge back. A subservient dog might be meek about it and yield quickly, but it´s not just going to sit there and look at you as if you´re crazy, like a cat might.
There was a relevant study done in England a few years ago, where they put GPS and cameras on all the cats in a village and tracked them constantly over some time.
Among other things, they found that cats timeshare territory. Cats in the same area, using the same territory, under normal circumstances there was never more than a single cat outside in the area, as they "took turns" to be out. If something upset the routine, there could be much staring and sometimes big fights, but normally they just stayed out of each others way, not seldomly sitting in a window staring at the others taking their turn.
So, effectively cats are socially antisocial, they can work and function together even despite essentially being loners, while dogs are so socially focused that they will seek out everyone and determine comparative standing, works well together as long as the "pack" has been properly determined, but barely at all if there is no clear ranking within a determined pack.
Ranma very much isn´t a loner, other than not by choice, and he´s ALWAYS out to show who´s #1(him!), except of course that he´s generally rather submissive outside of fighting, taking orders from his parents and the Tendo´s (among others) even when he really shouldn´t.
Totally dog behaviour.
A cat doesn´t get into fights "just because", and taking orders, hah...
And, of course, there are a lot more things to consider than just that one thing.
I wasn´t limiting myself to that, that was just the most obvious part that i thought of first.
There´s much more that could be said but i don´t have time(or geist) to get so convoluted now.