it seems you should care about at least some nonhuman animals.

I'm willing to entertain this possibility. I've recently been convinced that I should consider caring about dolphins and other similarly intelligent animals, possibly including pigs (so I might be willing to give up pork). I still don't care about fish or chickens. I don't think I can have a meaningful relationship with a fish or a chicken even in principle.

Moreover, it seems that insofar as you care about humans because they have certain psychological traits, you should care about any creature that has those traits. Since many animals have many of the traits that humans have, and some animals have those traits to a greater degree than some humans do, it seems you should care about at least some nonhuman animals.

Doesn't follow. If we imagine a personhood metric for animals evaluated over some reasonably large number of features, it might end up separating (most) humans from all nonhuman animals even if for each particular feature there exist some nonhuman animals that beat humans on it. There's no law of ethics saying that the parameter space has to be small.

It's not likely to be a clean separation, and there are almost certainly some exceptional specimens of H. sapiens that wouldn't stand up to such a metric, but -- although I can't speak for Qiaochu -- that's a bullet I'm willing to bite.

Since many animals have many of the traits that humans have, and some animals have those traits to a greater degree than some humans do, it seems you should care about at least some animals.

Does not follow, since an equally valid conclusion is that Qiaochu_Yuan should not-care about some humans (those that exhibit relevant traits less than some nonhuman animals). One person's modus ponens is etc.

Presumably mammals also exhibit more psychological similarity than non-mammals, and the same is probably true about East Asians relative to members of other races. What makes the psychological unity of mankind special?

I suspect that if you plotted all living beings by psychological similarity with Qiaochu_Yuan, there would be a much bigger gap between the -- [reminds himself about small children, people with advanced-stage Alzheimer's, etc.] never mind.