The issue is significantly more nuanced than that. Most people, and certainly most biologists and behavioral experts agree that there are certain animals that demonstrate sentience in a fashion at least analogous to the way humans think and feel. The great apes, and chimps, in particular, are among that rather rare group who share a significant number of emotional and cognitive traits with humans (little wonder, we're only separated by a few million years of evolution). So the idea here, so far as I understand it, is that those similarities are significant enough that chimps should enjoy, if not human rights, then at least some rights elevated from other far less human animals.
I tend to weigh on the side that sentient animals should receive protections similar to the protections we give to children or to adults deemed legally incompetent. That means they can't exercise many of the rights that we recognize adult humans have, but neither can they be wantonly exploited, physically or psychologically harmed.
But to pursue this in the courts is ludicrous. Personhood is fairly well defined in most, if not all, jurisdictions and it pretty much explicitly excludes anyone who isn't a member of H. sapiens. This is going to need to be something that is dealt with at the legislative level, and it's going to be a long fight.