Actually, it's cases like this that help define what constitutes rights and obligations for the plaintiff. To your example, corporations weren't automatically assumed to be people, but in the late 1800s cases were raised as to the validity of contracts held by corporations. The contracts were deemed valid, since the court ruled that people should not be deprived of their constitutional rights when they act collectively - therefore granting equal "rights" to corporations similar to individuals, in that narrowly defined context.
Corporations don't have wholesale similar rights to people. Currently a corporation cannot claim Fifth Amendment right against self-incrimination (that has been ruled upon).
However, in 2010 the Supreme Court upheld the rights of corporations to make political expenditures under the First Amendment. This was ruled upon by today's more conservative and pro-business Supreme Court, and many have said this is over-reaching. Which, if enough people and resources raise it as an issue, it may be overturned, or (more likely) specifically added as a constitutional amendment to abolish this rule.
This is why there is so much hand-wringing over new Supreme Court justices - they are 1/3 of the government's balance of power, and they serve a lifetime. If there is a president and majority party in Congress, they can influence the law of the land far beyond their tenure by appointing new judges that fit their political POV.