Earlier quoted context omitted.
At least for me, the problem is that making them completely equivalent in a legal sense has undesirable outcomes, like Citizens United. Having distinct terms allows for creating distinct, but potential overlapping sets of laws/privileges/rights. Using the same term makes it much harder to argue for distinctions in key areas
But they aren't completely equivalent. Natural persons can vote; juridical persons cannot. Natural persons have a constitutional right to avoid self-incrimination; juridical persons do not; etc. There's just a lot in common between the two, because it makes sense for there to be a lot in common. Citizen's United v FEC was a transparently terrible ruling, but it was in no way implied by the mere existence of corporate…
The entire cry of "corporations aren't people!" is based and a complete misunderstanding of what a legal person is. You've done a great job at explaining.
Unfortunately, there are a lot of people who willfully propagate these misunderstandings. Because by saying "of course corporations aren't people, and everybody knows this except those dumb ", it's an easy way to try to vilify the other side as dumb/evil. When the reality is that it's simply a tried-and-true necessary and useful legal concept, that virtually nobody but lawyers would even be familiar with in the first place, if it weren't for activists who thought it sounded scandalous.