The man was killed as he tried to enter a police station wielding a meat cleaver, on the first anniversary of the deadly Islamist attacks on the Charlie Hebdo satirical magazine.
An official account said the man, identified by a judicial source soon after the attack as Moroccan-born, shouted Allahu Akbar, (God is Greatest), and was wearing what turned out to be a fake suicide belt.
Molins told a French radio station the man may have given police a false identity some months ago. He also said a mobile phone found on the body was being examined and contained a German SIM card.
"I am not at all sure the identity he gave was real," Molins told France Inter radio on Friday.
A judicial source said on Thursday that the dead man was Ali Sallah, a Moroccan born in 1995 in Casablanca. He was homeless and known to police for theft in 2012 in the Var region of southern France.
Molins said authorities had established from fingerprints that the dead man identified himself as Sallah to police when they intercepted him last year. The name Ali Sallah was not known to intelligence services.
Molins said a sheet of paper found on the man's body gave a different name, and a Tunisian nationality.
A French police source told Reuters on Friday that a relative of a Tunisian man, Tarek Belkacem, had identified him as Thursday's assailant after seeing his photo on television and called police from Tunisia.
"It is a serious lead, but checks are under way to see if it really is this person," the source said. He added that the man in question was of Tunisian nationality and had emigrated to France.
Also on the sheet of paper was the Islamic State flag and a claim of allegiance to the militant group written in Arabic.
Islamic State, which controls swathes of Iraq and Syria, claimed responsibility for an attack in Paris on Nov. 13 in which 130 people died.
Molins said anti-terrorism authorities were working on 215 cases involving 711 individuals in France. Some 240 people had been taken in for questioning in connection with them.
He said about half the cases had reached the inquiry stage, and that authorities risked being overwhelmed because "since 2012 we have seen a doubling of these cases every year".
Belgian investigators said they believed explosives used in the November Paris attacks may have been made in an apartment in Brussels. It was rented under a false name and a fingerprint of a key fugitive was found.