Lewis Skelton, 31, has been identified by his family as the man Tasered and shot on Francis Street on Tuesday.
He later died in hospital. Officers had fired more than one shot at him, the Independent Police Complaints Commission (IPCC) said.
Mr Skelton's sister Laura Skelton said "my heart is breaking right now for my brother".
"He was a kind and caring person. What happened was out character. We are all heartbroken," she said.
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Neighbours described Mr Skelton as "mild mannered and polite".
Paul Mitten, who lived in the flat above Mr Skelton on Durham Street, said: "When I found out I just started crying because it just hit me for six knowing that what's happened over there was the lad who lived under me. He was quiet, placid, looked after his animals.
"I can't believe it. I can't get over it. I've not been able to sleep.
"He's always been polite to people, he's always said 'hi' when he's walked past... can't say a bad word about him."
The IPCC is investigating the shooting and said it was searching a flat in Durham Street as part of its inquiry.
Humberside Police said armed officers had been sent to the Holderness Road area at about 09:20 GMT following reports of a man carrying an axe.
The force said it had been responding to calls from members of the public.
Floral tributes have been left at the scene with one reading: "Fly high with the angels... we love you."
Another read: "Such a loss. What a lovely person he was. Words can't express the sadness we all feel, you didn't deserve this."
The sister of a man shot by police in Hull has described her brother's death as a "nightmare".