Not literally. While fire, police, and other services act "at your service", according to need, and essentially on demand, the bus has a schedule that -you- must conform to. If the bus company is subsidized with tax money, they -are- public servants in a broader sense.
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No. For any job to be a government job it can only be done by a government.
Since there are people who drive buses who do not work for the government,
bus driver cannot be classified as a government job.