Tesla shares plunged nearly 7 percent on Monday as investors registered dismay at Mr. Musk’s plans to form a third party and his intensifying feud with President Trump.

Musk has become the latest rich egomaniac to fall into this trap.

Weeks after the billionaire left his role at the Department of Government Efficiency amid his feud with President Trump, a small band of Musk loyalists is fighting to preserve the legacy—and power—of...