Osborne, from Cardiff, was arrested after pedestrians were targeted by a man driving a van near Finsbury Park Mosque in north London early on Monday.
Witnesses described hearing the van driver, who was detained by members of the public at the scene, shout: "I'm going to kill Muslims."
The man was initially arrested on suspicion of attempted murder but Scotland Yard said he was later arrested for the commission, preparation or instigation of terrorism including murder and attempted murder.
Residents near Cardiff said said they were "shocked" after seeing photographs of their neighbour being arrested in London.
Police are searching a residential address in Pentwyn, where Osborne is listed as living, in connection with the attack.
Five residents identified images of the man being arrested as their neighbour, Osborne, who according to reports is originally from Somerset.
Saleem Naema, 50, and his young son also said the man pictured was their neighbour.
"I can't believe it," Mr Naema, a taxi driver, said.
"I know him. I've lived here for five years, he was already living here when I moved in.
"If I ever needed anything he would come. I just can't believe that he did that. I am a Muslim."
Khadijeh Sherizi, who lives next door to Osborne, said: "It is definitely him.
"I saw him on the news and I thought 'oh my God' that is my neighbour.
"He has been so normal. He was in his kitchen yesterday afternoon singing with his kids.
"He was the dad of the family. He has kids. He lives next door. He seemed polite and pleasant to me.
"I just can't believe it."
Police said the attack happened while a man, who later died, was receiving first aid after being taken ill in Seven Sisters Road.
It is not clear whether there is any link between his death and the attack.
Deputy Assistant Commissioner Neil Basu said: "This is being treated as a terrorist incident and is being investigated by the counter terrorism command.
"The investigation is ongoing and we are working fast to know the full details of how and why this took place.
"All the victims were from the Muslim community and we will be deploying extra police patrols to reassure the public, especially those observing Ramadan."
The suspect, who is believed to have been from Weston-super-Mare, remains in custody at a south London police station.
The attacker, who is believed to have acted alone, struck as the area was busy with worshippers attending Ramadan night prayers at the mosque.
Eight people were taken to hospital, with one since discharged, while two others were treated at the scene.
All of the casualties were Muslims.
Earlier, Metropolitan Police Commissioner Cressida Dick said the incident was "quite clearly an attack on Muslims".
"We treat this as a terrorist attack," she said.
"Sadly we have suffered a number of attacks and very sad events over the last few weeks," she added.
Security Minister Ben Wallace confirmed the suspect was not known to the security services.
He told BBC Radio 4's World At One: "What I can say on this case is this individual, so far as we know at the moment, was not known to us, but we are aware of a rise in the far right."
Prime Minister Theresa May condemned the incident as "every bit as sickening" as the recent atrocities in London and Manchester.
Witnesses said the suspect was smiling and waving as he brought carnage to Seven Sisters Road, with video posted online showing him give a nonchalant wave as police put him in the back of their vehicle.
Imam Mohammed Mahmoud was hailed for his efforts to calm the chaotic situation before police arrived and was said to have used his body to shield the suspected terrorist from the fury of onlookers.
Images of the van showed it was rented from Pontyclun Van Hire in Pontyclun, near Cardiff.
Flowers have been laid near the scene, with one card reading: ''This is an attack on all Londoners - and on my community.''
Another read: ''I was so sorry to hear the news this morning. #NotInMyName''.
Another neighbour of Osborne on Glyn Rhosyn, in Pentwyn, said "It is definitely him."
The 30-year-old man, who declined to be named, said Osborne previously moved out of the address but had been living back with his partner and children there over the "past couple of months".
The man's partner added: "He is always arguing in the street with his missus. They usually shout and swear at each other."
Osborne is believed to have four children.