It’s conventional for American unions to have what we call “exclusive representation”, where the labor force within a determined “bargaining unit” picks them by majority vote and they then represent everyone in that unit. Non-exclusive unions exist and are protected (you may remember the news that a couple hundred Google workers formed one), but they’re generally not very powerful.
I worked as a labor organizer in the US and this response is closest to the correct understanding of the law. Workers have a right to engage in "concerted activity" for the purposes of improving working conditions. However, when workers organize a union it needs to be certified as having a right to negotiate terms and conditions of employment on behalf of the workers, which is almost always done through a union election.
In the U.S. this is an exclusive right—a "unit" of workers can only have one certified bargaining agent, and that union must represent all workers in the bargaining unit. It's also illegal for the employer to negotiate with any party other than the recognized union.
To my understanding, this is different from Europe and can seem strange to Europeans. Understanding how this framework came about historically can be useful. Before the National Labor Relations Act, workers would strike in order to gain recognition and the right for the union to negotiate on their behalf. The NLRA election framework was an attempt to bring about "labor peace" by introducing orderly elections instead.
This was generally supported by a certain segment of capital who could afford it and saw the cost-benefit as worth it, especially finance capital who were much more favorable to New Deal politics, and opposed by others. On the workers' side, this has undoubtedly made it easier to win recognition from the employer, but some see it as having sapped workers' militancy over the long run by trading away pitched battles that revealed sharp class antagonisms for a more orderly bureaucratic process.
