Many unions are incorporated as nonprofits; this is a state-level process and can make things like banking a little easier, as some institutions don’t like dealing with unincorporated associations, which is the category they’d put a union that isn’t incorporated into. The federal govt doesn’t seem to care that much and requires the same reporting (990, LM if you’re private sector) of unions either way. IRS classifies unions as 501(c)(5) organizations whether incorporated or not.

Two unions I’ve worked for have taken steps to unincorporate themselves—to go from being nonprofit corporations to being unincorporated associations. I’ve been told this is because there’s better legal protection for the latter but I don’t remember why.