A Syrian refugee, identified only as Mohammad A.R., sparked panic at the busy station on Monday after he started a fire in a McDonald's restaurant that left a 14-year-old girl with burns.
He then took a female employee hostage in a nearby pharmacy.
Heavily armed police opened fire on the 55-year-old suspect, who underwent emergency surgery and remains in a coma.
“Based on what we know so far, there are sufficient indications to suggest a radical Islamist background to this act,” federal prosecutors said in a statement after taking over the probe from local prosecutors.
In Germany, terrorist cases are dealt with by federal prosecutors. The accused made several demands by phone during the hostage negotiations, they said, including being allowed to travel to Syria to join the Islamic
State terror group.
Eyewitnesses have also said they heard the suspect say he belonged to IS and wanted to join the jihadist fighters in his native country.
The accused faces two counts of attempted murder and one charge of causing grievous bodily harm, the prosecutors added.
Cologne police said on Tuesday that Mohammed A.R. suffered from psychological problems and had a history of street offences.
He arrived in Germany from war-torn Syria in March 2015, and had since drawn police attention 13 times for offences including marijuana possession, theft, threatening behaviour and fraud.
Germany remains on high alert over the risk of a jihadist attack, having suffered several in recent years.
The bloodiest, claimed by IS, was a truck rampage through a Berlin Christmas market in December 2016 that left 12 people dead.