If corporations can be a person, can't labor unions and public school districts too?

"Personhood" conveys special protections to corporations.

But why can't "personhood" be legally extended to include Labor Unions, and public institutions
like school districts, affording them those same distinctions and protections?

Legally and politically speaking, is this doable?
Is it even desirable? Would there be special problems with this?

Anyone who is educated in Law is especially welcome to weigh-in here, along with anybody
else who has an opinion on this.

Thanks in advance for any responses.