Police searched the building in the western town of Recklinghausen but found no evidence of further attacks.
It is not clear if he was among the many migrants who arrived in Germany in 2015. He was reportedly arrested over a robbery in France in 2013.
He was shot dead on the anniversary of the 2015 Charlie Hebdo attacks.
The suspect was carrying a meat cleaver and wearing a dummy suicide vest.
A piece of paper found on his body "pledged allegiance" to so-called Islamic State (IS) and vowed revenge for French "attacks in Syria".
German police said he had drawn an IS symbol on the wall of a room in the shelter.
The man's exact origins are unclear. According to the German Welt am Sonntag newspaper, he has in various encounters with the French and German authorities claimed to be Moroccan, Tunisian, Syrian and Georgian.
It is not the first time IS attacks have been linked to suspects posing as refugees.
Two of the suicide bombers in the 13 November Paris attacks are believed to arrived on the Greek island of Leros in October and continued their journey through Europe with other migrants and refugees.
Some analysts have suggested that IS has encouraged a link between refugees and terrorism in order to foster hostility to refugees in Europe, although it is not known if the latest attack was carried out in co-ordination with the IS leadership in Iraq and Syria.
German police say a man who was shot dead as he attacked a police station in Paris on Thursday had been living in a shelter for asylum seekers in Germany.