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.