Authorities said the man, who was shot dead by anti- terror troops after Sunday's attack outside the southern city's main train station, had previously used a Tunisian passport under the name Ahmed H, 29.
But investigators are seeking to confirm his identity as the attacker -- who had a history of petty crime but was not on a jihadist watch list -- used seven aliases, anti-terror prosecutor Francois Molins told reporters.
The jihadist group's propaganda agency Amaq claimed the killer was one of its "soldiers", while a source close to the investigation told AFP no solid evidence linked him to IS.
The attack in France's second biggest city followed a string of stabbings around Europe claimed by or blamed on Islamist radicals.
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The man killed two 20-year-old cousins from the eastern city of Lyon. One was studying in Marseille and the other was visiting her for the weekend.
The attacker's fingerprints showed he had had seven brushes with the law since 2005 -- most recently when he was arrested last week in Lyon.
He presented the Tunisian passport to police, saying he was divorced, used "hard drugs", and had no fixed address.
The shoplifting charges were dropped for lack of evidence, and local authorities "were not able to take a decision to deport him," Molins added.
Police evacuated Marseille's ornate Saint Charles station after the attack, temporarily halting all train traffic on some of France's busiest lines.
French President Emmanuel Macron tweeted Sunday that he was "deeply angered by this barbaric act".
The attack comes as parliament prepares Tuesday to vote on a controversial anti-terror bill that transfers some of the exceptional powers granted to police under a 22-month-old state of emergency into national law.
France has been under a state of emergency since the IS gun and bomb attacks in Paris in November 2015 -- part of a string of jihadist assaults that have left more than 240 people dead over the past two years.
Knives have been the weapon of choice in a string of smaller-scale attacks in recent months, mainly targeting troops from the 7,000-strong Sentinelle anti-terror force set up to patrol the streets and vulnerable sites such as stations and tourist attractions.
In most cases, the attackers were shot dead at the start of their rampage, before they could kill others.
The Marseille attack came only days after IS released a recording of what it said was its leader Abu Bakr al-Baghdadi urging his followers to strike their enemies in the West.
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