A taxi, and particularly a bus driver, is the “chief” or “commander” of a small mobile unit with a lot of “horsepower.”
Such a driver is also responsible for the safety of several passengers. At least in New York City, this person is “in charge” not only of the vehicle, but everyone that is in/on it at any given time. A bus driver has the right to ask a passenger to move (to balance the vehicle) give a seat to a handicapped person, or do other things that make vehicle safer. Assaulting one carries extra penalties otherwise associated with assaulting a policemen.
Any Russian speaker in the city would understand why such terms are used.
In fact, Russian is a far better illustration of “master” for bus drivers than Cypriot Greek, which the question details suggest.
The Cypriot Greek term alluded to in details is mastros, which is cognate with master in English, and which derives either from Old French maistre or Venetian mestre. It has a cognate in Standard Greek mastoras, which grammatically points back to Byzantine Greek maistor, and ultimately Latin magister. (Of course, magister is also behind maistre and mestre and master.)
The thing is, words related to master don’t just mean “master”. Some of you will have noticed me address Michael Masiello as Magister, for example. That’s the (Mediaeval) Latin for “teacher”. A teacher is a master of their pupils, particularly in the old school way of schooling.
In Cypriot, the primary meaning of mastros is “boss”. A boss is a kind of master, but it’s the kind of master that makes sense under contemporary capitalism, rather than mediaeval feudalism.
There is a secondary meaning in Cypriot of mastros, which corresponds to the primary Greek meaning of mastoras. That meaning is “craftsperson, tradesperson”, and by extension “expert”. And it’s the friendly term with which you address a builder, a plumber, a carpenter, and so forth. It acknowledges their exercise of a practical skill.
That sense of course is hardly alien to English: we have master craftsmen and apprentices, we speak of someone being a master of their art or craft.
I don’t know for a fact why a bus driver in Cyprus is addressed as mastros. It could be, as in Russia, that you address him as “boss” because the vehicle, and your life, is entrusted in his hands. My own hunch is that you address him as “craftsman”, like you would any builder or plumber, out of deference to his professional exercise of skill.