(Not that there aren't plenty of those in Star Trek, too.)
Although, it appears the cat is named Lady.
Zoom in after the jump...
I referred to this drawing in my post on the episode "Breaking the Ice" and then I found it, so here it is.
It is hard to see, and my photo enhancing abilities are approaching nil, but there she is. Very clearly a cat wearing a bubble helmet in space (note the copious stars... and also a heart). You can't read it in this pic, but where my arrow is pointing, I believe I read it onscreen as saying "Lady", so I can only assume that is the space cat's name. Or maybe the feline space program uses ranks with titles like Lord and Lady. Maybe they model it after the Space Ren Faire.
The Vulcan is not labeled by name or gender, and does not have the benefit of a space helmet. He's a Vulcan, so it is hard to tell if he is bothered by that.