Fire is very dangerous, and dwarves have a very poor sense of that, so they can easily walk right up to it and get surrounded or succumb to smoke. A dwarf on fire will die unless the fire is doused (by water) and may die because of blood loss anyway (fire causes body fat on afflicted body parts to melt, which results in bleeding, which is staunched only by the supply of blood in that body part running out. It might be possible to staunch it in the hospital, but there isn't much time to get there before the body part has bled out). Dwarves can walk right up to fire, but the frequently fail to attack fire based creatures because they won't move that last tile into melee range because of the heat and just stand around while being attacked, which can result in some spectacular defeats.
FBs causing fire in the caverns sometimes cause critters to die due to that fire, although the FB frequently attacks them at the same time...
Fur should be more susceptible to fire than clothes, if anything.
Dragon fire will melt almost anything (including most magma safe materials) so a cat surviving dragon fire probably did so due to a legendary dodging ability. It might even be possible that you can dodge naturally occurring fire withing your own tile, which, if so, could explain why the other cats were unharmed.