An innovation in organized labour from around the turn of the 20th century known as the "industrial union" (as opposed to the "craft union") wherein the union seeks to organized all workers in a particular industry rather than a particular job. The idea is that this prevents workers of different jobs being pitted against one another, and provides more power as a strike could shut down an entire industry.

However, what is allowed is to require people who are hired to join the union, with the exception that if they do not want to join the union, they can opt to pay only "financial core" dues, to compensate the union for its work negotiating an agreement that the non-members supposedly benefit from too. The effect of this is that workers remain governed by collective bargaining, pay a lesser amount than members, and have no say in union leadership. They are also exempt from union discipline (ex fines for strikebreaking).

In some states, with so-called "right to work" laws, the employees who do not want to join the union pay nothing at all. This is actually not inherently a right of the states to make such laws, but an option they are delegated as part of the federal laws. After a recent SCOTUS ruling, all government employees can do this in any state.

In any case, what OP said is true re: our collective bargaining system is built around one union (whether or not everyone is required to pay dues) representing all workers, whose pay/benefits/other matters cannot vary from the union contract. Whether or not non-members must pay any dues

As to your point, the names for the old unions are largely historic vestiges. The "Big Labor" unions are massive conglomerates (just like the companies they rail against) that grow as big as they can. Ironically, the true hard-leftists like the IWW originally advocated for "One Big Union" as opposed to small, factional unions for each trade--but they would hardly be pleased with the highly-capitalist mammoth unions we have today.