The man was shot dead by French police on Thursday after he tried to storm the police station in northern Paris, brandishing a meat cleaver and wearing a fake suicide vest.
The assault took place exactly one year since the start of a series of jihadist attacks in France, beginning with the murder of 12 people at the Charlie Hebdo satirical magazine on January 7 2015.
Their statement said the man had lived at the shelter but gave no further details. No other attacks appeared to have been planned, it added.
A source close to the matter told AFP that the suspect had been registered as an asylum seeker.
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But French Interior Minister Bernard Cazeneuve cast doubt on the German claim.
"I cannot confirm this, quite simply because I am not at all sure that it is correct," Cazeneuve told France's iTele, and called on the media to exercise the "greatest care" in reporting the man's identity.
The head of North Rhine-Westphalia's criminal police service, Uwe Jacob said the suspect had travelled to Germany in 2013 for the first time from France, where he had lived illegally previously for five years.
He had gone under seven different identities and given at least three nationalities on separate occasions - Syrian, Moroccan or Georgian, Jacob said, according to national news agency DPA.
Welt am Sonntag said the man had drawn a symbol of the Islamic State organisation on the shelter's wall and had filed for asylum using the name Walid Salihi.
But French investigators said Friday the suspect appeared to have been identified by his family and was said to be a Tunisian named Tarek Belgacem.