Chekatt, who was born in Strasbourg in February 1989, was known to France‘s intelligence services as a potential security risk and authorities have launched a terror investigation.
BFM TV said the suspect was known to be part of radicalised networks in Strasbourg and was a “repeat offender” and a “delinquent”.
French authorities initially said the gunman had killed three people, but a French prosecutor said two people were confirmed dead while the third was brain dead.
The 29-year-old alleged gunman went on the run after the attack near Place Kleber, and France’s interior minister Christophe Castaner said 350 officers are hunting for him.
He exchanged fire with law enforcement as he “sowed terror”, Mr Castaner said.
The French government increased its security alert system Vigipirate to its highest level after the attack.
“Three hundred and fifty police and gendarmes are currently on the ground to apprehend the suspect, supported by two helicopters, the RAID [French anti-terror police], the BRI [anti-gang brigade] and the Sentinel force,” Mr Castaner said.
“The government has decided to move the security level to ‘Emergency Attack’ with stricter controls at the borders, and the implementation of reinforced controls on all the Christmas markets that are taking place in France to avoid the risk of copycat attacks.”
The suspected gunman, who has served time in prison in France and Germany for common law offences, was subject to a “fiche S”, literally an S card, which is used by police to flag someone who is considered a threat to national security. A fiche S (which stands for state security) allows for surveillance of a suspect, but is not grounds for arrest.
Others flagged under the system include Amedy Coulibaly, who killed four hostages at a siege in a Kosher supermarket in 2015, days after the Charlie Hebdo shooting.
The suspect’s home was also searched on Tuesday morning as part of investigations into a robbery. He was not home but grenades were found.
French military spokesman Col Patrik Steiger said the shooter did not seem to be aiming at soldiers patrolling in and around the market, but appeared to target civilians instead.
The suspect was shot and injured by soldiers guarding the Christmas market, but escaped in a taxi, BFM TV said.
He was said to have entered central Strasbourg via the Corbeau bridge to the south of the city centre, before heading to Rue des Orfevres, a popular shopping street close to the cathedral, where he opened fire at around 8pm local time (7pm GMT).
France has been hit in recent years with high-profile extremist attacks, including the coordinated attacks at multiple Paris locations that killed 130 people and wounded hundreds in November 2015, and a truck attack in Nice in 2016 which killed 86 and left hundreds injured.