BERLIN—Anis Amri, a Tunisian migrant whom authorities previously investigated for suspected terror ties and tried to deport, became Germany’s most wanted man as the new prime suspect in the capital’s deadly truck attack.

The revelation that the asylum seeker had been able to remain in Germany despite efforts to expel him stoked a furor over what many politicians called dangerous gaps in the country’s immigration policy and escalated the political crisis facing Chancellor Angela Merkel’s government.

The...