His unusual governing arrangement with President Trump is opening Republicans up to being yoked politically to Mr. Musk, who polls show is broadly unpopular.

In an attempt to motivate their voters and donors, a leading Democratic outside group recently launched ads in nearly two dozen House races to seize on Elon Musk’s role in slashing jobs in the federal government and paint GOP incumbents as willing to cut spending on health care for children and elderly to benefit billionaires.