The 45-year-old man, whose nationality was not given, was previously known to police, said authorities in Ruedesheim on the Rhine river in the central state of Hesse.
The woman, who had previously separated from her husband, was taken to hospital by helicopter, police said.
Two other people were injured in the fire in the shelter that houses 14 people.
The mass influx has forced Germany to accommodate the newcomers in hostels, sports halls and army barracks, where crowded conditions have at times sparked violent clashes.
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