Asylum office papers believed to belong to Amri, alleged to have links to the radical Islamist scene, were found in the cab of the 40-tonne lorry.
German prosecutors have issued a Europe-wide wanted notice for 24-year-old Anis Amri, offering a 100,000-euro ($104,000) reward for information leading to his arrest and warning he "could be violent and armed".
Asylum office papers believed to belong to Amri, alleged to have links to the radical Islamist scene, were found in the cab of the 40-tonne lorry that rammed through a crowded Christmas market in Berlin on Monday, killing 11.
The twelfth victim, the hijacked truck's Polish driver, was found shot in the cab.
Police Wednesday searched a refugee centre in Emmerich, western Germany, where Amri stayed a few months ago, as well as two apartments in Berlin, the media reported.
But as the Europe-wide manhunt intensified, questions were also raised about how the suspect had been able to avoid arrest and deportation despite being on the radar of several security agencies.
"The authorities had him in their crosshairs and he still managed to vanish," said Der Spiegel weekly on its website.
The Sueddeutsche Zeitung criticised police for wasting time focusing on a Pakistani suspect immediately after the truck assault, in what turned out to be a false lead.
"It took a while before the federal police turned to Amri as a suspect," it said.
The attack, Germany's deadliest in recent years, has been claimed by the Islamic State group.
Twenty-four people remain in hospital, 14 of whom were seriously injured.
Germany has boosted security measures following the carnage, beefing up the police presence at train stations, airports and at its borders with Poland and France.
In a revelation likely to stoke public anger, German officials said they had already been investigating Amri, suspecting he was planning an attack.
The interior minister of North Rhine-Westphalia state, Ralf Jaeger, said counter-terrorism officials had exchanged information about Amri, most recently in November, and a probe had been launched suspecting he was preparing "a serious act of violence against the state".
Berlin prosecutors said separately that Amri had been suspected of planning a burglary to raise cash to buy automatic weapons, "possibly to carry out an attack".
But after keeping tabs on him from March until September this year they failed to find evidence of the plot, learning only that Amri was a small-time drug dealer, and the surveillance was stopped.
In Tunisia, Amri's family expressed disbelief on hearing that Amri was wanted across Europe.
"I'm in shock, and can't believe it's him who committed this crime," his brother Abdelkader Amri told AFP.
But "if he's guilty, he deserves every condemnation. We reject terrorism and terrorists -- we have no dealings with terrorists."
Amri left Tunisia after the 2011 revolution and lived in Italy for three years, a Tunisian security source told AFP. Italian media said he served time in prison there for setting fire to a school.
He arrived in Germany in July 2015 but his application for asylum was rejected this June.
His deportation, however, got caught up in red tape with Tunisia, which long denied he was a citizen.
The apparent security failings in the case triggered fresh criticism of Chancellor Angela Merkel's liberal refugee policy, which has seen over a million people arrive since last year.
The record influx has fuelled support for the nationalist anti-migrant AfD party, which has accused Merkel of endangering the country.
"Nationwide, there are a large number of refugees about whom we don't know where they're from or what their names are. And that's a potential major security issue," said CDU member Klaus Bouillon, the interior minister of Saarland state.
Germany had until now been spared the devastating jihadist carnage that has struck neighbouring France and Belgium.
But it has suffered a spate of smaller attacks, including two attacks in July that left 15 people injured. Both were committed by asylum seekers and claimed by IS.
The Berlin Christmas market carnage evoked memories of the July 14 truck assault in the French Riviera city of Nice, where 86 people were killed by a Tunisian IS-sympathiser.