A German prosecutor said the man had run-ins with police regarding property damage, fraud and a hit-and-run, but all charges were dismissed each time.
While a motive has yet to be defined, cops said Sunday they "have no indications that there is a political background" for the attack in Muenster's old town section.
Officials said some of the 20 people injured were still in a life-threatening condition Sunday.
Muenster police president Hajo Kuhlisch said the man had four apartments and several cars, all of which were searched by police.
Chancellor Angela Merkel said she was "deeply shaken" by the incident and "everything possible will be done to determine what was behind this act and to help the victims".
Witnesses said people ran away screaming from the city square after the crash.
"We wish strength and fortitude to those who lost their loved ones and a speedy recovery to those injured by the heinous attack perpetrated in Munster, Germany", the statement said.
Armin Laschet, the governor of North Rhine-Westphalia state, where Muenster is located, visited the crash scene on Sunday.
He said, "People are calm, but really shocked", adding that people were scared and disbelieving that such an attack could happen in Muenster.
The attack came a year to the day of a truck attack in Stockholm in which a suspected Islamist militant sympathiser links killed five people.
"According to the current state of the investigation, the driver is probably a 48-year-old man from Muenster", said Chief Prosecutor of Muenster Martin Botzenhardt in the statement. "The investigations are being led under high pressure in all possible directions".
"The van is not at the crime scene any more, all kinds of objects have also been removed, waste of course, as well as evidence that we've found on the ground", police spokeswoman Susanne Dirkorte said.
Local media have identified the man as an industrial designer who had been suffering from psychological problems, but police would not confirm those details.
Firefighters of the city of Muenster clean the area in front of the traditional Grosser Kiepenkerl guesthouse where a man drove a van into a group of people sitting outside its restaurant in the old city centre of Muenster, Germany on April 8.
Muenster is a popular tourist destination with 300,000 inhabitants, known for its medieval old town, which was rebuilt after the massive destruction during World War II.