WASHINGTON — At a rally in Michigan a little over a week ago, President Trump assured his supporters that he had kept his promise to abolish the Affordable Care Act — even though Congress had failed to repeal the Obama-era health law.

“Essentially, we are getting rid of Obamacare,” Mr. Trump said, reminding a cheering crowd that the individual mandate that required most people to have health insurance or pay a penalty was scrapped as part of the Republican tax bill he signed into law last year. “Some people would say, essentially, we have gotten rid of it.”

But despite Mr. Trump’s longstanding desire to unwind the signature legislative achievement of his predecessor, many parts of the Affordable Care Act remain in place. And the Trump administration is even enforcing some of its provisions more aggressively than President Barack Obama did — a reality that has enraged business groups and Republicans in Congress who still want the law officially repealed.

While the individual mandate may be dead, the employer mandate — the requirement that many companies offer health insurance to their workers or pay a penalty — is very much alive. Under Mr. Trump, the Internal Revenue Service has been pursuing companies that fail to comply with the mandate and, according to the agency, was sending penalty notices to more than 30,000 businesses around the country.