“The fact that we have to work somebody from Wisconsin, where workers’ rights were decimated by Walker — that she’s even in play — says a lot,” D. Taylor, president of Unite Here, which represents about 10,000 workers employed by tribal casinos, said in an interview before the vote.

Ms. Baldwin’s office did not respond to a request for comment. But Pete Kirkham, a lobbyist who represents several tribes, noted that Indian sovereignty issues tended to loom large in Wisconsin. Some Democratic House members from the state have also supported the measure.

Supporters, including a coalition of business groups and Native American tribes, argued that the measure would simply restore the longstanding principle of tribal self-government, and that tribes shouldn’t be treated differently from state and municipal workers, who aren’t covered by federal labor law.

The argument is that a tribal government operating a casino is akin to a municipality generating revenue from a golf course.

“It merely seeks to treat tribal employers like any other public employer,” Dan Mahoney, executive director of the Native American Enterprise Initiative at the U.S. Chamber of Commerce, said in a statement. “This is an issue of parity. It is not an effort to deunionize but to clarify the paradigm under which unionizing could occur.”

At other times, supporters of the legislation have been more overtly antagonistic to federal labor rights. In 2015, Jefferson Keel, the lieutenant governor of the Chickasaw Nation, said one reason he opposed applying the National Labor Relations Act to tribal enterprises was that it protected the right to strike, potentially damaging casino operations.

“I would liken it to what happened with the air traffic controllers’ strike a number of years ago to this country,” he said at a congressional hearing. “We obviously are not on as large a scale, but that is the type of activity that would interfere with what we are doing.”