Harry Marchant was shot during Saturday's attack, which took place shortly before 3pm local time in Kraków's crowded Main Market Square.
The teenager, from York in north-eastern England, remains in hospital and is receiving treatment.
"We can't express how grateful we are to the first responders, medical team, and the people of Poland for their help and care for Harry," his family told the BBC.
York City FC, the football club Marchant supports, posted a message on Facebook on Tuesday, wishing him a speedy recovery and saying it hoped to see him back in the stands soon.
British media, including the BBC, The Scotsman and Sky News, reported on Wednesday that the British man killed in the attack was 32-year-old Scottish DJ James Campbell, from Prestwick, south-west of Glasgow.
The Scottish National Party's local branch expressed condolences to his family and friends. Police Scotland told The Scotsman it was supporting the victim's family and working on formal identification.
A 24-year-old Polish man, identified by prosecutors as Michał Oksiński, drove into the market square in Kraków on Saturday and opened fire on passers-by. Six people were killed and 16 injured, including the two police officers and the attacker himself.
The suspect was shot by police and taken into custody. He died in hospital early on Monday.
(sp)
Source: PAP