They aren't necessarily related but for lots of reasons animal rights is associated with utilitarianism. In particular, utilitarianism tends to recommend a much lower threshold of intelligence for an animal to be due our moral consideration- since the only requirement is experiencing pleasure/pain or having desires. Personhood is usually an epiphenomenal category in utilitarianism- referring to whatever class of entities we should be morally concerned with. It is often an essential category in deontology and it's confines much stricter- see Kantianism. Utilitarianism and expanding the sphere of moral concern are historically associated as well- Jeremy Bentham, Peter Singer etc. It is not unreasonably to infer from the popularity of utilitarianism here that animal rights is also popular.
Your reason is a good one too, though. And I'm not speaking from total ignorance here, either. I've been around these parts for a few years and I've seen plenty of upvoted comments about animal rights and had one or two discussions that bare on the subject. Someone is welcome to make a poll but I don't really think making the observation is worthy of downvotes.
For what it's worth, I don't care that much about animal rights; I think humans mostly care about humans; when they care about animals it's as a side effect of virtues whose primary purpose is to facilitate cooperation and peace between humans (and caring about animals is a good way of signaling those virtues).
(and I don't think intelligence and "personhood", whatever that is, have that much to do with each other.)
The October 2011 Scientific American has an editorial from its board of editors called "Ban chimp testing", that says: "In our view, the time has come to end biomedical experimentation on chimpanzees... Chimps should be used only in studies of major diseases and only when there is no other option." Much of the knowledge described in Luke's recent post on the cognitive science of rationality would have been impossible to acquire under such a ban.
I encourage you to write to Scientific American in favor of chimp testing. Some points that I plan to make:
I also encourage you to adopt a tone of moral outrage. Rather than taking the usual apologetic "we're so sorry, but we have to do this awful things in the name of science" tone, get indignant at the editors who intend to harm uncountable numbers of innocent people. For advanced writers, get indignant not just about harm, but about lost potential, pointing out the ways that our knowledge about how brains work can make our lives better, not just save us from disease.
You can comment on this here, but comments are AFAIK not printed in later issues as letters to the editor. Actual letters, or at least email, probably have more impact. You can't submit a letter to the editor through the website, because letters are magically different from things submitted on a website.
ADDED: Many people responded by claiming that banning chimp experimentation occupies some moral high ground. That is logically impossible.
To behave morally, you have to do two things:
1. Figure out, inherit, or otherwise acquire a set of moral goals are - let's say, for example, to maximize the sum over all individuals i of all species s of ws*[pleasure(s,i)-pain(s,i)].
2. Act in a way directed by those moral goals.
If you really cared about the suffering of sentient beings, you would also care about the suffering of humans, and you would realize that there's a tradeoff between the suffering of those experimented on, and of those who benefit, which is different for every experiment. That's what a moral decision is—deciding how to make a tradeoff of help and harm. People who call for a ban on chimp testing are really demanding we forbid (other) people from making moral judgements and taking moral actions. There are a wide range of laws and positions that could be argued to be moral. But just saying "We are incapable of making moral decisions, so we will ban moral decision-making" is not one of them.