After the 2004 elections, a Republican president, newly returned to office, decided the political moment was right to overhaul Social Security, making it more like private retirement plans in a bid to prevent it from going bankrupt in the future. Two years and a bruising political fight later, Democrats took back the House and Senate from Republicans.

Democrats now believe history may be about to repeat itself.

Elon Musk, the multibillionaire overseeing the Trump administration effort to drastically shrink government, has derided the nation’s most popular federal program as a sketchy pyramid scheme while pushing to close offices and eliminate thousands of jobs of those who administer the program. In doing so, he has touched a topic that has traditionally been known as the third rail of American politics. And Democrats, who increasingly regard Mr. Musk as a more opportune political target than even President Trump himself, have rushed to highlight what they consider to be a major political blunder.

“They don’t learn,” Senator Chuck Schumer of New York, the Democratic leader, said of Republicans as his party pounced on the issue. “Their biggest mistake was going after Social Security when George Bush was president. And now they are doing it again.”

Illustrating the emphasis Democrats intend to put on the subject, Senator Elissa Slotkin, the Michigan Democrat who delivered her party’s rebuttal to Mr. Trump’s congressional address on Tuesday night, hit on Social Security as well, using it to cast doubt on the president’s vows not to touch the federal retirement program.

“The president claims he won’t, but Elon Musk just called Social Security the biggest Ponzi scheme of all time,” she said in her remarks, quoting a social media post by Mr. Musk.