I think the (now deeply ingrained) confusion stems from a misconception surrounding the
Citizens United vs. FEC ruling, where the majority opinion in part cited campaign finance restrictions as a violation of First Amendment rights.
Since the Bill of Rights are protections of citizens rights, it has widely been misreported that this meant the SCOTUS ruling de jure recognized corporations to be citizens themselves.
This, of course, is not correct. The majority opinion in part says that "if the First Amendment has any force, it prohibits Congress from fining or jailing citizens, or associations of citizens, for simply engaging in political speech." The key phrase here being "associations of citizens", which the majority interpreted broadly to include corporations. This does not at all bestow personhood or citizenship upon corporations, legal or otherwise. It does afford corporations some protections as they are now legally recognized as "associations of citizens".