What's the difference between 我是老师 or 我当老师?
我是老师 = I am a teacher" It is completely correct

当 is a verb that means 'to be/ working as/ take on the role or position of' 

你当老师= you being a teacher

你来当老师 = you'll be a teacher

你去当老师 = you go and be a teacher

你当上了老师 = you've became a teacher


  The driver's question "你当老师?" (you being a teacher?) was a rhetorical one. That imply "It is hard to believe people as young as you would be a teacher." He could had asked "你是老师?" and the meaning would be the same.