UPDATE: I sent an e-mail to him about the naming thing. I tried to be as reasonable as I could, and to argue each of his points rationally, using words suggested by many of you.
He just responded to me with this tirade, starting off with insinuating that if I don't like his GM style (which, apparently is completely represented by the naming issues), I should probably leave the group. (I never suggested that I would leave the group if I didn't get my way -- in fact, I suggested that we compromise, or if not, I would accept his final ruling, but I wouldn't like it.)
Then he gave me some long-winded reason of why he liked things the way they were, and how his idea of light-hearted was different than mine. He cited all kinds of extreme character names that he saw once he started gaming with people other than his core group of friends. (My animal names were nowhere near as funny or weird.)
THEN he proceeded to tell me what kinds of weird animal names he would accept. I may as well say now what names I had used thus far: Mittens the lioness (from the Simpsons, "my cats name is Mittens"), and Marshmallow the bear (from having fed bears marshmallows at the zoo when I was a toddler -- yes, I'm that old). Here are the names he would accept: "Patches, Rex, One-Eye, T-Bone, Blue, Caspar, Dusty, Lefty, Scratches, or Oddball are all good examples of fine pet names. It's when I imagine the name belonging to a little old ladies sweater wearing dog or cat that I have a problem with." Oddly, Rex, one of the examples I cited earlier in this post, would be acceptable. And Mittens is acceptable. So, to me, his idea of silly is a bit incomprehensible. I just don't see the distinction between "Patches" and "Marshmallow."
He then said that he thought the difference between his idea of light-hearted and mine was because I am a woman. *rolling my eyes*
Lastly, he said he would not compromise, and, the part that really stuck in my throat, was when he said, "if I'm going to spend hours upon hours preparing a game so I can DM it for other people, my enjoyment needs to come before theirs." This, I don't get. I'm not just some pawn in his world, there to play for his pleasure. We are all in this together. Doesn't it need to be fun for everyone?
His response seemed to me to be overly hostile and selfish. He always said to bring up rules problems out of game so that it doesn't slow down the game. So I did. And now I'm treated with obstinance, sexism, irrationality, and selfishness. Maybe I do need to walk over a name.