Chekatt was shot in the arm during a gunfight with police but managed to escape in a taxi, boasting of the attack to the driver, who was forced to drop him in another part of the city.
Chekatt had 27 convictions in France, Germany and Switzerland, and authorities believe he became radicalised while in jail. He was flagged for extremism and was on a watch list. Strasbourg is on the border with Germany, where the suspect was convicted in 2016 of breaking into a dental practice and a pharmacy in two towns.
Two of his sisters in Paris were also questioned on Thursday and one of their homes was searched, a judicial source said.
Prosecutors had opened a terror investigation into the attack.
For several hours swathes of the city were under lockdown.
One Italian was reported to be among the wounded.
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Five of the wounded were in serious condition, the prefecture of the Strasbourg region said.
Meanwhile, authorities revised the death toll to three on Thursday after a person who had been kept alive while brain dead succumbed to his injuries.
The attack comes nearly exactly two years after the Berlin Christmas attack that saw failed Tunisian asylum seeker Anis Amri ram a truck through the city's market killing a dozen people and injuring dozens more.
He was killed after a two-day manhunt that prompted a curfew in the eastern French city near the German border and forced the country to raise its national security threat level.
Antonio Tajani said that the legislature "will not be intimidated by terrorist or criminal attacks".
In January 2015, 17 people were killed and another 22 wounded in an attack by al-Qaida jihadists on the headquarters of the satirical magazine Charlie Hebdo and a kosher market in Paris.
The market was to remain closed Thursday, city officials said.
Manhunt continues after fatal Strasbourg Christmas market shooting; suspect identified
In response to Tuesday's shooting, the government chose to take the country's attack risk up a level on the official threat index. Nunez said it was believed that the attacker had been injured by security forces Tuesday night but that could not be confirmed.
Chekatt allegedly shouted "God is great!" in Arabic and sprayed gunfire from a security zone near the Christmas market.
While some of the movement's representatives have said they are open to halting the protests to negotiate with the government, others have said its concessions are not enough, and have vowed to protest again in Paris this weekend.
No group has claimed responsibility for the Strasbourg attack, but social media accounts used by Daesh sympathizers have celebrated the killings.
This version corrects the spelling of the suspect's last name to Chekatt, not Chekkat.
"Considering the target, his way of operating, his profile and the testimonies of those who heard him yell "Allahu Akbar", the anti-terrorist police have been called into action, " Heitz explained.
Police seized a grenade, a rifle and knives in the operation, Heitz said, and began guarding the apartment building.
The office said the man is a member of the "entourage" of main suspect Cherif Chekatt, 29, but not a family member.
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Hundreds of police and elite anti-terror units in France were on Wednesday hunting for the gunman who was known to police and thought to be an religious extremist. "You can just tell", said one.