This just isn’t true. At a constitutional level, for instance, corporations have no Fifth Amendment right against self-incrimination. Lower courts have extended this principle even to one-person corporations. At a state level, corporations (whether straight or gay) are denied the right to marry or adopt a child.
It would be accurate to say that the law extends many of the same rights and protections to corporations as it does to individuals. It would be more accurate to say that individuals retain many rights and protections when they associate as corporations. But to say simply that those rights are “the same” is not supportable.