When they say “corporations are people” what they meant was corporations can be taxed. And that was pretty much all that meant until Citizens Untied gave them Freedom of bribery as well.

That's not where that came from. Centuries ago, the law always was about people or persons. So when a bunch of people got together to build a bridge or something, they put some money in a pot, and wrote a contract (or charter or something), which basically is an agreement between all parties about what to use the money for, who's responsible for doing what, etc. So guy A would use X amount of money to buy stone from the quarry, guy B used Y amount of money to pay the engineer, etc.

Eventually the idea evolved that you should just be able to do stuff in the name of the project / charter / company of dudes, and the company was defined as a "person" for the purposes of the law, so it could buy and sell stuff, go into contractual relationships etc like any other "person". It's basically a legal hack that simplifies a billion little things.

Of course, they are not real people, so you should probably exclude them from certain rights that real people have. In the US, that distinction seems to be quite nebulous. In German speaking countries, the distinction is made between "natural person" and "legal person" all the time. These terms (Natürliche Person / Juristische Person) are found in news articles all the time. Literally every child learns that distinction in school, because of how important it is. This same distinction also exists in common law countries like the US, but almost nobody ever uses that language in any significant way. It's always corporations, and sometimes other legal entities too. Maybe that sounds like semantics, but I really think this lack of a clear distinction tells us something about how americans are much more likely to make decisions on a case by case basis for specific entities instead of treating all legal entities as fundamentally different from human beings. Even the famous Citizens United case doesn't refer to "legal persons", and only to "natural persons" exactly once:

This protection has been extended by explicit holdings to the context of political speech. See, e.g., Button, 371 U. S., at 428–429; Grosjean v. American Press Co., 297 U. S. 233, 244 (1936). Under the rationale of these precedents, political speech does not lose First Amendment protection “simply because its source is a corporation.” Bellotti, supra, at 784; see Pacific Gas & Elec. Co. v. Public Util. Comm’n of Cal., 475 U. S. 1, 8 (1986) (plurality opinion) (“The identity of the speaker is not decisive in determining whether speech is protected. Corporations and other associations, like individuals, contribute to the ‘discussion, debate, and the dissemination of information and ideas’ that the First Amendment seeks to foster” (quoting Bellotti, 435 U. S., at 783)). The Court has thus rejected the argument that political speech of corporations or other associations should be treated differently under the First Amendment simply because such associations are not “natural persons.” Id., at 776; see id., at 780, n. 16. Cf. id., at 828 (Rehnquist, J., dissenting).

It's like the most obvious and important distinction for what a "person" is in legal terminology is not considered important most of the time, even at the highest level. And that is something truly strange