The article notes several countries that have declared certain animals to be non-human persons, and the notion of corporate person is widely recognized.

If "we" refers to the USA, there isn't the political will to extend these rights to animals such as elephants or primates for many reasons (eg., Voter apathy, special interest groups that use animal testing, zoos, etc.).

Even if one sovereign decided to grant some animals personhood, that wouldn't necessarily be recognized outside that jurisdiction (e.g., two nations, state vs. federal govt.), so they would likely only be a "person" in specific contexts.