That's not how language works. We're also all animals but not animals at the same time. Context and culture affect the meaning of words.
The article doesn't really provide any details as it's only a few lines long.
The word "primate" in itself is not a slur but it can be used as a slur, as you said, depending on the context, however not much context was given in the article.
Do you have a source for that? I couldn't find this in the article but maybe I missed it.