Aaron Lewis, 26 and of Toxteth, was shot three times outside Anna Jung's Chippy in Grosvenor Road, Wavertree, at about 19:00 GMT on Wednesday.
Police said the "cold-blooded, targeted attack" was related to organised crime.
His family said his death had "broken their hearts" and they "wanted justice" but "no more gun death".
A post-mortem examination revealed Mr Lewis died of multiple gunshot wounds.
On Wednesday evening a gun was found in a Renault van which was found burnt out near Brookdale Road and Ferndale Road in Wavertree.
It will now be forensically examined to establish if it can be linked to the murder, police said.
Mr Lewis, who was known to police, was the son of kickboxing champion Alfie Lewis.
His tribute read: "Last night my oldest son was shot dead. It has broken my family's heart; we loved him from the very first day he was born.
"We do not want revenge because that would mean meaningless loss of another life.
"We will leave it in the hands of the police who gave us great support and for that we want to thank them dearly.
"Let the law take its course. We want justice. No more gun death."
A 19-year-old man went to hospital with a gunshot wound to his upper body shortly after Mr Lewis was shot.
His condition is not thought to be life-threatening and detectives believe he had been with Mr Lewis.
On Thursday police staged a show of strength in the city, with a high-profile operation in some of the areas worst affected by a spike in gun and gang crime.
Det Supt Richie Davies said children as young as 10 were being used as gun-runners by gangs in Liverpool.
He said vulnerable youngsters were "being coerced" into storing and minding firearms for criminal groups.
Merseyside Police said gun crime had increased in recent months after falling year on year from 2013 to 2016.
Since April, 78 firearms have been discharged, 18 more than in the whole of the previous 12 months.
The force has sent letters to residents to inform them that "non-suspicion" search powers are being used against those suspected of having offensive weapons.
The powers, under Section 60 of the Criminal Justice and Public Order Act 1994, are used to "tackle serious violent crime", the letter said.
It said the legislation was not currently widely used and the "parameters are reviewed on a daily basis".
In the last financial year there were fewer than 1,000 of these "anticipation of violence" stops in England and Wales.
Detectives investigating the fatal shooting of a man in Liverpool are examining a handgun found in a burnt-out car near the scene.