Corporations are not granted the same rights as natural persons - they're granted some of the rights of natural persons (and some of those make sense while other are dumb).
Preventing corporations from operating as persons when contributing to political campaigns would be an absolute win for representation... Citizens United was a trash ruling by trash people.
Preventing corporations from operating as persons when interacting with the legal system would massively complicate our legal code and probably lead to a lot of awful edge cases where they could dodge liability for their actions. They are generally immune to manslaughter and some industries have dumb political carve outs (like firearms manufacturers having immunity to most product liability) but people's ability to sue, i.e. a paper mill after getting luekimeia relies on corporate personhood in our current system.
When we talk about "corporations shouldn't be people" we (I think it's fair to be generally inclusive here) are talking about their ability to donate to politicians and political causes.