As President Trump delivered his inaugural address in 2017, a slight woman with feathered gray hair sat listening, bundled in a hooded white parka in a fenced-off V.I.P. section. Her name was Rosemary T. Vrablic. She was a managing director at Deutsche Bank and one of the reasons Mr. Trump had just taken the oath of office.

It was a moment of celebration — and a moment of worry for Ms. Vrablic’s employer.

Mr. Trump and Deutsche Bank were deeply entwined, their symbiotic bond born of necessity and ambition on both sides: a real estate mogul made toxic by polarizing rhetoric and a pattern of defaults, and a bank with intractable financial problems and a history of misconduct.