Police are trying to trace the driver after the pensioner, who has not been named, suffered head and leg injuries on a trip home from Tesco on Oystermouth Road, Swansea.
It has been linked to a similar incident when Eirwen Ferguson, 85, was injured in a taxi from the same store.
The council said it was also investigating.
The daughter of the 100-year-old woman said she was hurt in the four-mile (6.4km) drive.
"He was chopping and changing lanes and came to a sudden halt because there was a car in front of him," she said.
"She was sitting behind the driver and catapulted forward and hit her head on the top of the seat. His seat moved backwards and she screamed. She injured her leg."
She claimed her mother knocked her head a second time after the driver "hit the kerb", adding she required medical treatment for bruising on her head and a haematoma on her injured leg.
A South Wales Police spokesman said: "All efforts to trace the taxi and the driver have proved unsuccessful. Anyone who knows the identity of the taxi or the driver is asked to call police."
A 100-year-old woman has been injured in a "fast and furious" taxi drive home which left her "terrified".