Kailash Chander is understood to have lost control and driven on to the pavement before hitting at least four lampposts and crashing into the city centre store on Saturday evening.
Chander, a father of two and former Labour councillor and former town Mayor of Leamington Spa in Warwickshire county, had worked with Stagecoach Midlands for 42 years, a colleague said, describing him as "a polite, kind man who has done a lot for the community," Metro.Co.Uk reported.
An eight-year-old boy, a passenger on the top deck of the bus, and the woman thought to be in her 70s, a pedestrian, both died at the scene, West Midlands Police have said.
A nine-year-old girl is also seriously injured in hospital, while six others, including the bus driver, were hurt.
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The bus collided with stationary cars before hitting the Sainsbury's store. Police have not commented on what caused the crash on Saturday.
"We hope to carry out a swift investigation and I would like to thank people who have already come forward who witnessed the collision.