Seven of the dead were teenagers - two Turks, two Germans, a Hungarian, a Greek and a Kosovan.
At least 35 people were also wounded during Sonboly's attack, which began at a McDonald's outlet and ended with him turning his 9mm Glock pistol on himself.
Bavaria's top security official on Sunday urged a constitutional change to allow the country's military to be able to be deployed in support of police during attacks like Friday night's deadly rampage at a Munich mall, while Germany's vice chancellor proposed even stricter controls on firearms.
Heimberger also said the gunman likely purchased his illegal weapon online, through a website trafficking illegal weapons hosted on the dark web.
Sonboly had spent two months at an inpatient facility in 2015 and was treated afterward as an outpatient, Steinkraus-Koch said.
The teenager had been planning the attack for a year, police say.
According to police, 58 bullet casings were found at the shooting scene and 57 of them were from the gunman's weapon, while one was from a police weapon.
Heimberger said there is still no evidence of any political motivation to the crime, nor that the shooter killed specific victims.
Police have been probing claims that the killer felt bullied by his peers and that he may have been inspired by Norwegian mass murderer Anders Breivik, who killed 77 people exactly five years before the Munich shootings.
Documents found in his home confirmed that he suffered from mental illness, including depression and anxiety, Munich prosecutor Thomas Steinkraus-Koch said Sunday.
"He completely occupied himself with this act of rampage".
In a post on a chatroom website, which has since been taken down, a former classmate wrote of how schoolchildren would harrass and bully Sonboly.
Munich police chief Hubertus Andrae ruled out an Islamist militant link to the attack.
The solo shooting spree saw nine people murdered and 27 others injured - 10 of whom are in a critical condition - before he took his own life.
The Afghan youth was questioned after he contacted police following the shooting on Friday, the statement said.
Police were investigating findings suggesting the Munich gunman invited people to a fast food restaurant at the mall via the Facebook account, Heimberger told the news conference.