Discussions of an immortal soul aside, those animals may suffer - but they don’t have a conscious, reflective “self” that knows that they are suffering.
I quite agree. However, what she actually said is, "When it comes to having a central nervous system, and the ability to feel pain, hunger, and thirst," these animals are much like us. Which is of course quite true.
The list of other elements we have in common is almost endless: We're all mammals, we're vertabrates, we all contain DNA, we're all omnivores, etc., etc., etc.
Newkirk's "argument" here exhibits several types of logical fallacies. The most obvious being the non sequitur. She produces these facts, and then "reasons" from these facts to the idea that these animals should have legal rights. But her conclusion does not necessarily follow from these facts.