Simon Nellist, from Wolli Creek, was swimming at Little Bay when he was attacked just after 4.30pm on Wednesday.
It was Sydney's first fatal shark attack since 1963 at Sugarloaf Bay in Middle Harbour.
Nellist, who was born in the UK, was remembered by his family as being a "very talented photographer".
"Simon was a gentle, kind and wonderful human being. He was a cherished fiancee, son, brother, uncle and friend," his family told the BBC.
"Simon was funny, compassionate and always had time for people. He had a rare gift of instantly being able to connect with others, gaining their trust and respect."
Nellist's family said he was a "proud Cornishman" who had made Australia his home.
"Simon had a great passion for nature and the sea, as well as being a very talented photographer," they said.
The family said they would "miss him terribly".
Nellist, who life savers said sustained "catastrophic" injuries in the attack, swam at Little Bay every day.
Nellist's social media is filled with underwater pictures, showing him as an avid ocean lover.
The dive instructor was part of Scuba Diving Social Club in Sydney's south.
All beaches in Sydney's east and south re-opened on Friday after being closed on Thursday after the attack as authorities searched for the shark and more remains.
Local residents attended a vigil for Nellist at The Coast Chapel in Little Bay on Friday.
NSW Premier Dominic Perrottet last week sent his condolences to the family.
"Our hearts are with you at this difficult time, it's a reminder I guess to us all of the fragility of life," Mr Perrottet said.
Police and the Department of Primary Industries have the power to destroy the animal if it is deemed an extreme risk to public safety, but it is understood the preference would be to tag and track the shark.
But marine scientist Vanessa Pirotta last week said the shark that attacked Nellist might never be caught.
Pirotta told the ABC it was likely the shark, estimated to be 3 metres long, had already left the area and "we will never see it again".
"Shark incidences like this are rare and uncommon," the Macquarie University expert said.
"Now, it's not every day that we see an animal of this size off our beaches, but it is possible.
"These animals are capable of large geographical movements and we will see them from time to time."
The shark could have been drawn to the area by currents, or because it was chasing prey or seeking warmer water, Dr Pirotta speculated.
"Or simply this animal may have just been passing through the area and unfortunately, in this case, has had a negative interaction with a human," she said.