Michael Cohen, President Donald Trump’s personal attorney, used the same Delaware limited-liability company in two secret deals relating to alleged sexual encounters involving his clients, according to people familiar with the matter.

Venture capitalist Elliott Broidy paid an initial installment of $62,500 to the company, Essential Consultants LLC, as part of Mr. Cohen’s $250,000 total fee for negotiating a nondisclosure agreement related to Mr. Broidy’s affair with a former Playboy model who alleged he had impregnated her,...