Tucked into a stack of budget bills Gov. Tim Walz of Minnesota signed into law last May was a measure that went largely unnoticed: Public schools would be required to provide free menstrual products to all students who needed them, beginning in the fourth grade.

The measure was spearheaded by teenagers like Elif Ozturk, who had listened to classmates in the restroom fret over not having a tampon, and Maarit Mattson, who began carrying extra pads in middle school, as a gesture of support for friends uncomfortable confiding in their parents about the start of their periods.

“It isn’t part of a political agenda,” Ms. Mattson, 15, said. “It was to make students feel safe.”

The new law took effect on Jan. 1 this year, with little sign of public opposition, and Minnesota joined about half of all states in passing legislation to improve access to menstrual products in public schools.

But when Vice President Kamala Harris named Mr. Walz as her running mate for the presidency this month, conservative commentators seized on one phrase in Minnesota’s law, which guarantees access to the products for “all menstruating students.”

Labeling Mr. Walz “Tampon Tim,” opponents including former President Donald J. Trump have accused the governor of a heavy-handed campaign to place tampons and pads in boys’ restrooms for transgender students, tying it to a broader conservative backlash over Mr. Walz’s support for L.G.B.T.Q. rights.