Not true, exactly. For legal purposes, corporations fall into a class called "juridic persons". They have some rights that natural persons have. For example, they have the rights to free speech (look up the Citizens United SCOTUS decision).
I have to disagree. A juridical person is a legal entity which achieves legal personhood through incorporation. The important point is that it is created by the state, as opposed to being created by God. As our constitution acknowledges rights as above the reach of the state, rights therefore cannot be granted BY the state. Thus corporate "rights" are a term of art indicating the only thing the state CAN grant - limited privileges. That's why the laws are so whack, because the state can changed those privileges at any time.
Juridical persons were actually first created by the Catholic Church through Canon Law, to represent an aggregate of human beings being treated as a single person. This seems to be the conceptual framework used to address the Church itself. But translated into statutory law in the US, it directly conflicts with the founding relationship between the state and the people. Therefore it is assigned corporate status.
The use of the word "rights" for corporations is a common, but severely misleading abuse of the mechanism of terms of art. The courts get away with it by pretending they are writing only for attorneys "well versed in the art," (since these are virtually all appellate rulings, as appellate hearings can only be filed and argued by attorneys). This single misuse of this particular word has served to hide the truth about this issue for a very long time - to tremendous detriment. It is literally THE answer to the common American cry of "what happened to my rights?!"