A union is recognized by the NLRB and has certain statutory entitlements regarding strikes and the ability for employers to fire striking workers. "Labor movements," writ large, do not require those statutory entitlements. Modern unions can be understood in terms of their ability to collectively bargain without fear of direct reprisal in terms of employment. Otherwise, collective bargaining has always been an option (it's just humans acting in concert), and long predated the concept of "unions."

So you would call a labor party a worker union? Labor seems to be able to organize just fine regardless if there are unions or not, as long as they are allowed to vote.

Would you say that a labor party is a union? I wouldn't. A labor party is a labor movement though. If you call labor parties unions and say we should be thankful to them, then I agree, but that is totally different from workplace unions.