A.P. Moller-Maersk A/S’s third-quarter profit plunged 43% as weak freight rates and low oil prices continued to batter the Danish shipping-and-oil company’s results.

Maersk’s shares sank more than 7% in Copenhagen on Wednesday. Investors fear that the shipping industry—the company’s biggest market—remains caught in its deepest down-cycle in 30 years.

Maersk...