Think of it more as you have a top-level protocol in a piece of software. It's super flexible, designed for many diverse use cases. You write a whole pile of software around the top-level protocol "Person". Then, you realize you need to extend the software to support a new kind of thing, the "River". You now have a few choices.
Option 1: Massive refactor. Create a new top-level thing called "LegalEntity" to which both "Person" and "River" conform. Then, you have to update literally every single thing in the entire app to support both, and handle both appropriately.
Option 2: There's enough flexibility in "Person" than you can just make "River" conform to "Person" and you don't have to refactor anything at all.
The rights between a human, a corporation and a river might be different, but the fact that they all have rights is what unites them and why they are "persons". I believe it is a legacy when only human beings had rights. But the legal system got abstracted and we wanted to give rights to non-human entities.
Also, keep in mind that human beings had different rights as well. Some humans didn't even have rights and were not considered legal persons.
Colloquially, we use person as a synonym for human, but in law, person is anything with rights.
I would guess that the issue here is some mix of your not understanding the legal nuance here (as you allude to) and the law around it itself being incoherent. I don't claim to understand the laws here myself, but then the legal concept of a person is very tortured anyway.
Person vs Property.
https://en.wikipedia.org/wiki/Civil_forfeiture_in_the_United...
It's the category of things that have legally cognizable interests, which is arguably the most important category for law.
It's named “persons” because persons in the boring common sense were originally the only members of he category.
It works, and wise people don’t seek to fix things that are not actually broken.
In the US a certain set of rights are given to "persons" that should really be given to "legal entities", I've never had a conclusive answer as to why but I'm just going to guess it was a lack of foresight. Over time there have been judicial rulings, national laws, local laws, and other fun stuff that have slowly corrected or clarified the definition of a legal "person" that should apply to all things labeled "persons". Some of these you may or may not agree with (maybe only "natural persons" should be able to participate in the political system and such) but the general approach of clarifying the term even if the term isn't perfect is a pretty good approach.
That this meaning is unknown to people outside the profession is not necessarily an issue. It is the job of a good legal professional to make sure we never need know whilst they sort the minutiae between themselves.
We don't expect scientists to restrict their terminologies within their own fields to terms with which laypeople are familiar, equally so for those of us in computer science and engineering, so why expect it of another field?
The high priesthood preserved Latin, it's true, concealing the true nature of the religion from the tongue of its faithful parishioners in order to maintain unearned control; this is not the case for Law English as lawyers and judges have nothing to gain by maintaining their own manner of speaking since their pronouncements and judges do need to be explained, in plain English, to the involved parties and the public who have no need of the knowledge of legal minutiae. These 'translations' are relatively easily scrutinised even by non-professionals and a public that is increasingly literate in the law.